🧠:[<think>Alright, let's tackle this geometry problem. Hmm, okay, so we have a circle k with center S, and two diameters AB and CD. That means AB and CD both pass through the center S, right? So AB and CD are straight lines that cross at S, the center. Since they're diameters, the length of AB and CD should be equal to twice the radius of the circle. The angle BAD is given as 28 degrees. So point A is one end of diameter AB, and point D is another point on the circle. Since AB is a diameter, angle ADB should be a right angle because any triangle inscribed in a semicircle is a right triangle. Wait, but angle BAD is 28 degrees, not angle ADB. Let me visualize this. Points A and B are endpoints of a diameter, so if we look at triangle ABD, with AB as the diameter, then D must lie somewhere on the circle such that angle BAD is 28 degrees. Next, there's a circle centered at A that passes through S, the center of circle k. This circle intersects circle k at points E and F, with D and F on the same side of line AB. So, the circle centered at A passing through S means that the radius of this circle is equal to AS. Since S is the center of circle k, AS is the distance from A to the center, which is the radius of circle k. Wait, no. If AB is a diameter of circle k, then the radius is half of AB, so AS is equal to the radius. Therefore, the circle centered at A with radius AS would pass through S and also through point B? Wait, AB is a diameter, so AB is twice the radius. So AS is the radius, so the circle centered at A with radius AS would pass through S and another point on AB. But since AB is a diameter, S is the midpoint, so AS is half of AB, so the circle centered at A with radius AS would have a radius equal to half of AB. Therefore, starting from A, moving a distance AS (which is half AB) along AB would reach point S. But then, the circle centered at A with radius AS would pass through S and would also extend to other points not on AB. This circle intersects the original circle k at points E and F. Since both circles intersect, their intersection points E and F lie on both circles. The problem mentions that D and F are on the same side of line AB. So, once we find points E and F, F is the one that's on the same side as D with respect to AB. We need to determine the measure of angle CFS. Point C is another endpoint of diameter CD. Since CD is a diameter, C is diametrically opposite D. So, if we can figure out the positions of points C, F, and S, then we can find angle CFS. Let me try to sketch this mentally. Circle k with center S. Diameters AB and CD. Since AB and CD are diameters, they intersect at S. The angle BAD is 28 degrees. Let me recall that in a circle, the angle subtended by an arc at the center is twice the angle subtended at the circumference. So maybe there's a way to relate angle BAD to some central angles. Since AB is a diameter, angle ADB is 90 degrees, as I thought earlier. But angle BAD is 28 degrees. So in triangle ABD, we have a right triangle at D, with angle at A being 28 degrees, so angle at B would be 62 degrees (since angles in a triangle add up to 180). Wait, angle at B is angle ABD. So angle ABD is 62 degrees. But maybe that's not immediately useful. Let's think about the circle centered at A passing through S. Since S is the center of circle k, the distance from A to S is equal to the radius of circle k. Therefore, the circle centered at A with radius AS is congruent to circle k? Wait, no. Because if AS is the radius of circle k, then circle centered at A with radius AS would have the same radius as circle k. So, circle k has center S and radius AS, and the other circle has center A and radius AS. Therefore, these two circles intersect each other. The distance between their centers is AS, since S is the center of circle k and A is the center of the other circle. The distance between centers is AS, and both have radius AS. So, the distance between centers is equal to the radius. Therefore, the two circles intersect at two points E and F such that triangle AES and AFS are equilateral? Wait, maybe. Because if you have two circles with centers separated by a distance equal to their radius, then the points of intersection form equilateral triangles with the centers. Let me recall: if two circles with radius r have centers separated by r, then the triangle formed by the two centers and one of the intersection points is an equilateral triangle, since all sides are equal to r. Therefore, points E and F would form equilateral triangles with A and S. So, angles at E and F would be 60 degrees. Hmm, but does that apply here?Yes, because the distance between centers A and S is AS, which is equal to the radius of both circles. Therefore, triangle AFS is equilateral, so angle FAS is 60 degrees. Similarly, triangle AES is equilateral. So points E and F are located 60 degrees away from S as viewed from A? Wait, maybe not exactly. Let's think. If we have two circles intersecting, both with radius AS, and centers A and S, then the intersection points E and F lie such that AE = AS = SE and AF = AS = SF. So triangles AES and AFS are both equilateral. Therefore, angles at S, like angle ASE and angle ASF, are 60 degrees. Therefore, points E and F are located on circle k such that the arcs from S to E and S to F are 60 degrees each. Wait, because in circle k, points E and F are located 60 degrees away from A? Not exactly. Wait, if triangle AES is equilateral, then angle EAS is 60 degrees. Since point E is on circle k, which is centered at S, the position of E relative to S can be determined. Hold on, maybe coordinate geometry would help here. Let me try to assign coordinates to simplify.Let’s set the center S at the origin (0,0). Let’s assume circle k has radius 1 for simplicity, so AS = 1 (since S is the center and A is on the circle, so radius is 1). Then, point A is at (1,0), since we can place it on the x-axis. Since AB is a diameter, point B is at (-1,0). Now, angle BAD is 28 degrees. Point D is on circle k. Let's find the coordinates of point D. Since angle BAD is 28 degrees, with points B at (-1,0), A at (1,0), and D somewhere on the circle. The angle BAD is the angle at point A between points B, A, and D. So in triangle BAD, vertex at A, angle 28 degrees. Since A is at (1,0), BA is the vector from A to B, which is (-2,0), and DA is the vector from A to D, which we need to determine. Wait, maybe using vectors is complicating. Alternatively, since angle BAD is 28 degrees, and points B, A, D are on the circle, perhaps we can use the arc corresponding to that angle. But angle BAD is an inscribed angle. The measure of angle BAD is half the measure of the arc BD. So, angle BAD = 28 degrees implies that arc BD is 56 degrees. Therefore, the arc from B to D is 56 degrees. Since the circle has 360 degrees, the arc BD is 56 degrees. But since AB is a diameter, the arc from A to B is 180 degrees. If arc BD is 56 degrees, then starting from B, moving 56 degrees along the circumference towards D. Depending on the direction, but since we have diameters AB and CD, perhaps CD is perpendicular to AB? Not necessarily, unless specified. Wait, the problem says AB and CD are diameters, but doesn't specify their orientation other than angle BAD being 28 degrees. Wait, since angle BAD is 28 degrees, and AB is a diameter, the position of D can be determined. Let me think. If we have point A at (1,0) and point B at (-1,0), then angle BAD is 28 degrees. So point D is somewhere on the circle such that when you connect A to D, the angle between BA and DA is 28 degrees. In coordinate terms, the vector from A to B is (-2,0), and the vector from A to D is (D_x - 1, D_y - 0). The angle between these two vectors is 28 degrees. The angle between vector AB and AD is 28 degrees. Since AB is along the negative x-axis from A, the direction from A to B is 180 degrees. The angle between AB and AD is 28 degrees, so AD is making a 28-degree angle above the line AB (since D and F are on the same side of AB, but wait, D is on the circle, so depending on the orientation). Wait, actually, the problem states that D and F are on the same side of line AB. So when we determine point D, we need to ensure it's on the same side as F. But perhaps we can figure this out with coordinates. Let me calculate coordinates of point D. Since angle BAD is 28 degrees, and AB is along the x-axis from A(1,0) to B(-1,0). So the line AB is the x-axis. The angle between BA and DA is 28 degrees, so DA is making a 28-degree angle above the x-axis (assuming D is above AB; since F is on the same side as D, and the problem specifies D and F are on the same side, we can assume D is above AB). Therefore, point D is located at an angle of 28 degrees above the x-axis from point A. However, since A is at (1,0), moving 28 degrees from the x-axis towards the upper half-plane. Wait, but angle BAD is at point A, so the direction from A to D is 28 degrees above the line AB (which is the x-axis). Wait, AB is from A to B along the x-axis towards the left. So angle BAD being 28 degrees would mean that AD is 28 degrees above BA. Since BA is pointing to the left along the x-axis, 28 degrees above BA would be 28 degrees into the upper half-plane (second quadrant from A's perspective). But since point D is on the circle centered at S(0,0) with radius 1, we need to find its coordinates. Alternatively, perhaps using polar coordinates. Let's consider point A at (1,0). The angle BAD is 28 degrees. Since BA is the line from A to B, which is towards the left along the x-axis. The angle BAD is measured from BA to AD, which would be 28 degrees upwards from BA. So from point A, looking towards B (left along the x-axis), turning 28 degrees upwards (counterclockwise) gives the direction of AD. Therefore, the direction of AD is 180 - 28 = 152 degrees from the positive x-axis. Wait, no. If from point A, the direction towards B is 180 degrees (straight left), then turning 28 degrees upwards (counterclockwise) from that direction would be 180 - 28 = 152 degrees from the positive x-axis. So the line AD makes an angle of 152 degrees with the positive x-axis. But point D is on the circle centered at the origin with radius 1. So we can parametrize point D as (cos θ, sin θ). The line AD goes from (1,0) to (cos θ, sin θ). The angle between BA (which is along the negative x-axis) and AD is 28 degrees. To find θ, we can use the formula for the angle between two vectors. The vector BA is (-2, 0), and the vector DA is (cos θ - 1, sin θ - 0) = (cos θ - 1, sin θ). The angle between BA and DA is 28 degrees. The formula for the angle φ between vectors u and v is:cos φ = (u · v) / (|u| |v|)Here, u = BA = (-2, 0), so |u| = 2.v = DA = (cos θ - 1, sin θ), so |v| = sqrt((cos θ - 1)^2 + (sin θ)^2) = sqrt(cos²θ - 2 cos θ + 1 + sin²θ) = sqrt(2 - 2 cos θ) = 2 sin(θ/2) [using the identity sqrt(2 - 2 cos θ) = 2 sin(θ/2)]The dot product u · v = (-2)(cos θ - 1) + 0*(sin θ) = -2 cos θ + 2Therefore,cos φ = (-2 cos θ + 2) / (2 * 2 sin(θ/2)) ) Wait, wait. Let's compute this step by step.cos φ = [(-2)(cos θ - 1) + 0*sin θ] / (|BA| |DA|)Which is [ -2 cos θ + 2 ] / (2 * |DA| )But |DA| is sqrt( (cos θ - 1)^2 + sin²θ ) = sqrt( cos²θ - 2 cos θ + 1 + sin²θ ) = sqrt( 2 - 2 cos θ )Therefore,cos φ = (2 - 2 cos θ) / (2 * sqrt(2 - 2 cos θ)) )Simplify numerator and denominator:cos φ = [2(1 - cos θ)] / [2 sqrt(2 - 2 cos θ)] ) = (1 - cos θ) / sqrt(2 - 2 cos θ)Note that 1 - cos θ = 2 sin²(θ/2) and 2 - 2 cos θ = 4 sin²(θ/2), so sqrt(2 - 2 cos θ) = 2 sin(θ/2)Therefore,cos φ = (2 sin²(θ/2)) / (2 sin(θ/2)) ) = sin(θ/2)Given that φ = 28 degrees, so cos φ = cos 28° = sin(θ/2)Therefore,sin(θ/2) = cos 28°But cos 28° = sin(62°), since sin(90° - x) = cos x. Therefore,sin(θ/2) = sin(62°)Which implies θ/2 = 62° or θ/2 = 180° - 62° = 118°, so θ = 124° or θ = 236°But since D is on the same side of AB as F, and we assumed earlier that D is above AB (since F is on that side), θ would be 124°, which is in the upper half of the circle. θ = 236° would be in the lower half, which is the other side of AB. Therefore, θ = 124°, so point D is at (cos 124°, sin 124°)Calculating cos 124° and sin 124°, 124° is in the second quadrant. Cos 124° = -cos(56°), sin 124° = sin(56°). So coordinates of D are (-cos56°, sin56°). Approximately, cos56° ≈ 0.5592, sin56° ≈ 0.8290, so D is approximately (-0.5592, 0.8290).Now, we need to find points E and F where the circle centered at A (1,0) with radius AS (which is the distance from A to S). Since S is at (0,0), the distance AS is sqrt( (1-0)^2 + (0-0)^2 ) = 1. So the circle centered at A(1,0) with radius 1. Equation of circle k (center S(0,0) radius 1): x² + y² = 1. Equation of circle centered at A(1,0) radius 1: (x - 1)^2 + y^2 = 1.To find points E and F, we solve these two equations:1. x² + y² = 12. (x - 1)^2 + y^2 = 1Subtracting equation 1 from equation 2:(x - 1)^2 + y^2 - x² - y² = 1 - 1 => (x² - 2x + 1) - x² = 0 => -2x + 1 = 0 => x = 1/2Therefore, the points of intersection lie on x = 1/2. Plugging back into equation 1:(1/2)^2 + y² = 1 => 1/4 + y² = 1 => y² = 3/4 => y = ±√(3)/2 ≈ ±0.8660Therefore, points E and F are at (1/2, √3/2) and (1/2, -√3/2). Now, the problem states that D and F are on the same side of line AB. Line AB is the x-axis. Since D has a y-coordinate of sin124° ≈ 0.8290 (positive), F must be the intersection point with positive y-coordinate. Therefore, F is (1/2, √3/2) and E is (1/2, -√3/2). Wait, but the problem says "D and F are on the same side of the line AB". Since AB is the x-axis, D is above the x-axis, so F must also be above. Therefore, F is (1/2, √3/2) and E is (1/2, -√3/2).So point F is at (1/2, √3/2). Now, we need to determine angle CFS. Point C is the other endpoint of diameter CD. Since D is at (cos124°, sin124°), which is (-cos56°, sin56°), the diameter CD means that C is diametrically opposite D, so C is at (-cos124°, -sin124°) = (cos56°, -sin56°). Because if D is at (x,y), then C is (-x,-y). Wait, is that correct? Since CD is a diameter, so S is the midpoint. Therefore, coordinates of C can be found by rotating D 180 degrees around S. So if D is (x,y), then C is (-x,-y). Therefore, since D is (cos124°, sin124°), C is (cos(124° + 180°), sin(124° + 180°)) = (cos304°, sin304°) = (cos(-56°), sin(-56°)) = (cos56°, -sin56°). So point C is at (cos56°, -sin56°). Approximately, cos56° ≈ 0.5592, sin56° ≈ 0.8290, so C is approximately (0.5592, -0.8290).Point S is at (0,0), so we need to find angle CFS, which is the angle at point F between points C, F, and S. So angle at F: ∠CFS.To find this angle, we can use coordinates to compute vectors FC and FS, then find the angle between them.First, coordinates of points:C: (cos56°, -sin56°) ≈ (0.5592, -0.8290)F: (0.5, √3/2) ≈ (0.5, 0.8660)S: (0,0)Vector FC = C - F = (cos56° - 0.5, -sin56° - √3/2)Vector FS = S - F = (-0.5, -√3/2)Wait, angle at F is between points C, F, S, so the angle is formed by FC and FS. So we need vectors FC and FS. Vector FC goes from F to C: (C_x - F_x, C_y - F_y) = (cos56° - 0.5, -sin56° - √3/2)Vector FS goes from F to S: (S_x - F_x, S_y - F_y) = (-0.5, -√3/2)We can compute the angle between vectors FC and FS using the dot product formula:cosθ = (FC · FS) / (|FC| |FS|)First, calculate FC · FS:= (cos56° - 0.5)(-0.5) + (-sin56° - √3/2)(-√3/2)= -0.5 cos56° + 0.25 + (sin56° * √3/2 + (√3/2)^2 )Simplify term by term:First term: -0.5 cos56°Second term: +0.25Third term: + (√3/2) sin56°Fourth term: + ( (√3/2)^2 ) = 3/4So altogether:= (-0.5 cos56° + 0.25) + (√3/2 sin56° + 3/4 )Combine constants: 0.25 + 3/4 = 1So:= -0.5 cos56° + √3/2 sin56° + 1Now, let's compute this value numerically.cos56° ≈ 0.5592sin56° ≈ 0.8290√3 ≈ 1.732So:-0.5 * 0.5592 ≈ -0.2796√3/2 * 0.8290 ≈ (0.8660) * 0.8290 ≈ 0.7175So total:-0.2796 + 0.7175 + 1 ≈ 1.4379Now, compute |FC| and |FS|.First, |FS|:FS is from F(0.5, √3/2) to S(0,0), so the distance is sqrt( (-0.5)^2 + (-√3/2)^2 ) = sqrt(0.25 + 0.75) = sqrt(1) = 1. So |FS| = 1.|FC|: distance from F(0.5, √3/2 ≈ 0.8660) to C(cos56° ≈ 0.5592, -sin56° ≈ -0.8290)So:Δx = 0.5592 - 0.5 ≈ 0.0592Δy = -0.8290 - 0.8660 ≈ -1.695So |FC| ≈ sqrt( (0.0592)^2 + (-1.695)^2 ) ≈ sqrt(0.0035 + 2.8720) ≈ sqrt(2.8755) ≈ 1.696Therefore, cosθ ≈ 1.4379 / (1.696 * 1) ≈ 1.4379 / 1.696 ≈ 0.848Therefore, θ ≈ arccos(0.848) ≈ 32 degrees.Wait, but let's check if this makes sense. Alternatively, maybe there's a geometric interpretation without coordinates.Alternatively, since we know coordinates, maybe using complex numbers or geometric properties.But let's see, angle CFS is approximately 32 degrees. However, considering the problem gives angle BAD as 28 degrees, which is close to 30 degrees, and we ended up with 32 degrees. Maybe the exact value is 30 degrees or something else. Wait, perhaps there's a mistake in my approximate calculations. Let me attempt exact calculation.Alternatively, let's work with exact expressions.Given:cos56° = sin34°, but perhaps not helpful.Wait, angle CFS is formed at point F(0.5, √3/2). Let's note that point F is one of the intersections of the two circles. Since the circle centered at A(1,0) with radius 1 intersects circle k at F(0.5, √3/2). Let me recall that triangle AFS is equilateral, as both AF and AS are radii of the circle centered at A, and FS is also a radius of circle k. Wait, AF is equal to AS = 1, and FS is the distance from F to S, which is 1, since F is on circle k. Therefore, triangle AFS is equilateral, so all angles are 60 degrees. Therefore, angle AFS is 60 degrees. But we need angle CFS. Maybe there's a relationship between points C, F, S, and other points.Since CD is a diameter, CS is the line from C to S to D. Since S is the center, CS is a radius. Wait, CD is a diameter, so CS is half of that, but CS is a radius. Wait, no. If CD is a diameter, then CS is a radius. Wait, CD is a diameter, so the length of CD is 2*radius, so CS (from C to S) is radius. Wait, in our coordinate system, the radius is 1, so CS is 1 unit. Similarly, CF is the distance from C to F. Let's compute CF exactly.Point C is (cos56°, -sin56°), point F is (0.5, √3/2). So the distance CF is sqrt( (cos56° - 0.5)^2 + (-sin56° - √3/2)^2 )But this seems complicated. Alternatively, let's consider triangle CFS. We have points C, F, S with coordinates:C: (cos56°, -sin56°)F: (0.5, √3/2)S: (0,0)We can compute the lengths of sides:FS: 1 (as before)CS: distance from C to S: sqrt(cos²56° + sin²56°) = 1 (since C is on circle k)CF: distance from C to F: sqrt( (0.5 - cos56°)^2 + (√3/2 + sin56°)^2 )But maybe using the Law of Cosines on triangle CFS. In triangle CFS, we know two sides: CS = 1, FS = 1, and we want angle at F (angle CFS). Wait, no. Wait, angle at F is angle CFS, which is the angle between FC and FS. The sides are:- FC: distance from F to C- FS: distance from F to S = 1- CS: distance from C to S = 1Wait, triangle CFS has sides:CS = 1FS = 1CF: ?So, in triangle CFS, two sides are 1 (CS and FS), and the angle at F is angle CFS. But unless CF is also 1, which would make it equilateral, but CF is not equal to 1. Wait, unless points C, F, and S form an equilateral triangle, but from coordinates, point C is (cos56°, -sin56°) ≈ (0.5592, -0.8290), point F is (0.5, 0.8660), so CF is not equal to 1. Wait, perhaps using the Law of Cosines in triangle CFS:We have sides CS = 1, FS = 1, and angle at S: angle CSF. Wait, no. Wait, Law of Cosines needs one angle and two sides or three sides. If we can find CF, then we can use the Law of Cosines to find angle CFS.Alternatively, using vector approach as I tried earlier.Wait, in the coordinate system, angle CFS is the angle at F between points C, F, S. So vectors FC and FS.From earlier calculation, the dot product FC · FS ≈ 1.4379, |FC| ≈ 1.696, |FS| = 1. So cosθ ≈ 0.848, leading to θ ≈ 32 degrees. However, 0.848 is approximately cos(32°), since cos30°≈0.866, cos32°≈0.848. So angle ≈32 degrees. But is there an exact value?Wait, perhaps there's a smarter way. Let me think geometrically.Since triangle AFS is equilateral, angle AFS = 60°. We need angle CFS. Points A, F, S are all related. Let's see.Point C is diametrically opposite D. Since angle BAD =28°, and D is at 124°, point C is at 124° +180°= 304°, which is equivalent to -56°. Now, point F is at (0.5, √3/2), which is 60° from the x-axis. So in polar coordinates, F is at 60°, radius 1. Point C is at 304°, which is equivalent to -56°, so in polar coordinates, 304° is 360° -56°, so it's in the fourth quadrant. Now, angle CFS is the angle between vectors FC and FS. Since S is the origin, and F is at (0.5, √3/2), vector FS is (-0.5, -√3/2). Vector FC is C - F = (cos304° - 0.5, sin304° - √3/2) = (cos(-56°) - 0.5, sin(-56°) - √3/2) = (cos56° - 0.5, -sin56° - √3/2)But maybe there's a symmetry or another approach. Alternatively, since point F is at 60°, and point C is at 304°, the angle between them from the center S is 304° -60°=244°, but that's the reflex angle. The smaller angle between them is 360° -244°=116°. But that's the angle at the center between C and F. Then, since angle at the center is 116°, the inscribed angle would be half that, 58°, but angle CFS is not an inscribed angle over that arc. Wait, angle CFS is at point F, so it's an angle formed by points C, F, S. To relate this to the arcs, we can use the inscribed angle theorem. The angle at F would be half the measure of the arc CS that does not contain F. Wait, no. The inscribed angle theorem states that the measure of an angle formed by two chords in a circle is equal to half the sum of the measures of the intercepted arcs. In this case, angle CFS intercepts arcs CS and... wait, need to be precise. Angle CFS is formed by chords FC and FS. The vertex is at F, which is on the circumference. Therefore, the measure of angle CFS is equal to half the measure of the arc from C to S that does not contain F. Wait, no, the angle formed by two chords at a point is equal to half the sum of the measures of the intercepted arcs. Wait, the rule is: the measure of an angle formed by two chords intersecting at a point on the circumference is half the sum of the measures of the intercepted arcs. But angle CFS is formed by chords CF and SF. So the intercepted arcs are CS and... wait, maybe I need to think again. If two chords intersect at a point on the circumference, the angle is half the sum of the intercepted arcs. But here, angle at F intercepts arcs CS and... another arc? Wait, maybe not. Wait, if we have angle CFS, which is at point F, formed by points C, F, S. The intercepted arcs are the arcs that are opposite the angle. In this case, the angle at F intercepts the arc from C to S passing through the other side of the circle. Since the center angle between C and S is 116°, as previously calculated, but that's the larger arc. The smaller arc between C and S is 360° -116°=244°, which doesn't make sense. Wait, no. From point C (304°) to point S (which is the center, but S is not on the circumference. Wait, maybe I confused S with another point. Wait, S is the center, not on the circumference. Therefore, angle CFS is not an inscribed angle in the circle, because S is the center. Therefore, the angle is not subject to the inscribed angle theorem. Therefore, maybe my previous approach with coordinates is the way to go. But let's try another approach. Since triangle AFS is equilateral, then angle SAF=60°, and SA=AF=FS=1. Point A is at (1,0), S is at (0,0), F is at (0.5, √3/2). Point D is at (cos124°, sin124°), which we can write as (cos(180°-56°), sin(180°-56°)) = (-cos56°, sin56°). Therefore, coordinates of D are (-cos56°, sin56°). Point C is diametrically opposite D, so C is (cos56°, -sin56°). We can consider triangle CFS. Points C(cos56°, -sin56°), F(0.5, √3/2), S(0,0). Let's compute vectors:Vector FC = C - F = (cos56° - 0.5, -sin56° - √3/2)Vector FS = S - F = (-0.5, -√3/2)We need the angle between FC and FS. Let's calculate the dot product:FC · FS = (cos56° - 0.5)(-0.5) + (-sin56° - √3/2)(-√3/2)= -0.5 cos56° + 0.25 + sin56° * (√3/2) + ( (√3/2)^2 )= -0.5 cos56° + 0.25 + (√3/2 sin56°) + 3/4Combine constants: 0.25 + 0.75 = 1So:= -0.5 cos56° + (√3/2 sin56°) + 1Now, let's recognize that √3/2 = cos30° and 0.5 = sin30°, so:= -sin30° cos56° + cos30° sin56° + 1This resembles the sine of difference identity:sin(A - B) = sinA cosB - cosA sinBSo:-sin30° cos56° + cos30° sin56° = sin56° cos30° - cos56° sin30° = sin(56° - 30°) = sin26°Therefore, FC · FS = sin26° + 1So,cosθ = (sin26° + 1) / ( |FC| |FS| )But |FS| is 1, as before. Now, compute |FC|:|FC| = sqrt( (cos56° - 0.5)^2 + (-sin56° - √3/2)^2 )Expand the terms:= sqrt( cos²56° - cos56° + 0.25 + sin²56° + √3 sin56° + 3/4 )Combine cos²56 + sin²56 = 1= sqrt( 1 - cos56° + 0.25 + √3 sin56° + 0.75 )Simplify constants: 0.25 + 0.75 = 1= sqrt( 1 - cos56° + 1 + √3 sin56° )= sqrt( 2 - cos56° + √3 sin56° )Again, let's express this in terms of sine and cosine. Let's factor out a 2:= sqrt( 2 + ( -cos56° + √3 sin56° ) )Note that -cos56° + √3 sin56° is similar to 2 sin(56° - 30°) = 2 sin26°, but let's check:We can write -cos56° + √3 sin56° as 2*(-0.5 cos56° + (√3/2) sin56°)Which is 2*(sin60° sin56° - cos60° cos56° ) Because sin60°=√3/2, cos60°=0.5So:= 2*( - (cos60° cos56° + sin60° sin56° ) )= -2*cos(60° - 56° ) = -2*cos4°Wait, using the identity cos(A + B) = cosA cosB - sinA sinB, but here we have:-cos56° + √3 sin56° = 2*(-0.5 cos56° + (√3/2) sin56° ) = 2*( sin60° sin56° - cos60° cos56° )= -2*( cos60° cos56° - sin60° sin56° ) = -2*cos(60° +56° ) = -2*cos116°Wait, cos(A + B) = cosA cosB - sinA sinB, so:cos60° cos56° - sin60° sin56° = cos(60° +56° ) = cos116°Therefore:-cos56° + √3 sin56° = -2 cos116°Therefore,|FC| = sqrt( 2 - 2 cos116° ) = sqrt( 2(1 - cos116° ) )Using the identity 1 - cosθ = 2 sin²(θ/2), we get:= sqrt( 2 * 2 sin²(116°/2 ) ) = sqrt(4 sin²58° ) = 2 sin58°Therefore, |FC| = 2 sin58°So, going back to cosθ = (sin26° + 1 ) / ( |FC| |FS| ) = (sin26° + 1 ) / (2 sin58° * 1 ) Now, let's compute sin26° + 1:We can write sin26° + 1 = 1 + sin26°And 2 sin58° = 2 sin(60° - 2° ) ≈ but maybe there's an identity.Alternatively, use co-functions:Note that sin58° = cos32°, since sin(90° - θ ) = cosθ.So,cosθ = (1 + sin26° ) / (2 cos32° )Let’s compute numerator and denominator:1 + sin26° = 1 + sin26°2 cos32° = 2 cos32°We can use the identity:1 + sinα = 2 sin(45° + α/2 ) cos(45° - α/2 )Let’s apply this with α =26°:1 + sin26° = 2 sin(45° +13° ) cos(45° -13° ) = 2 sin58° cos32°Therefore,cosθ = [2 sin58° cos32° ] / [2 cos32° ] = sin58°Therefore,cosθ = sin58° But sin58° = cos(32° ), since sinθ = cos(90° - θ )Therefore,cosθ = cos32° Therefore, θ =32°, since angle between vectors must be between 0° and 180°, and cos32° is positive, so angle is 32°.Therefore, angle CFS is 32 degrees.But wait, the problem statement specifies angle BAD is 28°, and our answer is 32°, which is 4° larger. This seems a bit arbitrary, but perhaps it's correct. Let me verify once more.Alternatively, maybe there's a miscalculation in the trigonometric identities. Let me double-check.We had:FC · FS = sin26° +1|FC| |FS| = 2 sin58° *1=2 sin58°Then, using the identity 1 + sin26° = 2 sin58° cos32°, how?Wait, let's verify:Using the identity:sinA + sinB = 2 sin( (A+B)/2 ) cos( (A-B)/2 )But we have 1 + sin26°. Let me write 1 as sin90°, so:sin90° + sin26° = 2 sin( (90+26)/2 ) cos( (90 -26)/2 ) = 2 sin58° cos32°Yes, exactly. So sin90° + sin26° = 2 sin58° cos32°, so 1 + sin26° = 2 sin58° cos32°. Therefore, the numerator is 2 sin58° cos32°, denominator is 2 sin58°, cancels to cos32°, so cosθ = cos32°, so θ=32°. Therefore, angle CFS is 32 degrees.But the problem is given in exact terms, angle BAD is 28°, which is not a standard angle, but the answer came out to 32°, which is 28° +4°. Is there a general relationship here?Alternatively, perhaps there's a different approach using properties of circles and angles.Recall that angle at F, CFS, is the angle we need. Point F is on both circles, the original circle k and the circle centered at A passing through S. Since AF = AS =1, triangle AFS is equilateral, so angle SAF is60°, as we had earlier.Given that angle BAD =28°, and we have point D, which is on the circle k such that angle BAD=28°, and point C is diametrically opposite D.Perhaps there is a cyclic quadrilateral or some other relationship. Let's consider quadrilateral CFDS or something similar.Alternatively, since S is the center, CS and DS are radii. Also, FS is a radius. Maybe triangle FCS has some special properties.Alternatively, notice that angle CFS is equal to angle CAS plus some other angle. Wait, but not sure.Alternatively, since we know the coordinates and the angle came out to 32°, which is 28° +4°, but that might not mean anything. Given that the calculation led us to 32°, which is exact based on the steps, but maybe the problem expects an answer of 28°, but that contradicts the computation.Wait, let's check once again with exact angles:We have:cosθ = sin26° +1 divided by 2 sin58°, which simplifies via identities to cos32°, leading to θ=32°. Therefore, the answer should be 32 degrees.But let's confirm with approximate degree measures:sin26° ≈0.4384,1 +0.4384=1.4384,2 sin58°≈2*0.8480=1.6960,1.4384 /1.6960≈0.848,cos32≈0.848,yes, so θ=32°. Therefore, angle CFS is32 degrees.But the problem might expect the answer in an exact form, possibly relating to the given angle of28°, perhaps double or something. But according to the calculation, it's32°. Hmm.Wait, maybe there's a different approach where angle CFS is 2 times angle BAD. 2*28=56°, but our answer was32°, so that doesn't fit. Alternatively, complementary angle, 90-28=62°, no. Alternatively, 60°-28°=32°. Ah! 60° minus28° is32°, which is our answer. Given that triangle AFS is equilateral, with angles of60°, and angle BAD is28°, which is part of the geometry, the difference between60° and28° is32°, which might be the answer. This seems plausible. Therefore, angle CFS=60° -28°=32°.This could be the case. Therefore, the answer is32 degrees, which is 60° -28°, due to the equilateral triangle and the given angle.Therefore, the measure of angle CFS is32°, which should be boxed as the final answer.Final Answerboxed{32^circ}</think>Given a circle ( k ) with center ( S ) and diameters ( overline{AB} ) and ( overline{CD} ), where ( angle BAD = 28^circ ). A circle centered at ( A ) passing through ( S ) intersects ( k ) at points ( E ) and ( F ), with ( D ) and ( F ) on the same side of line ( AB ). We need to determine ( angle CFS ).1. Circle Geometry and Intersection: - The circle centered at ( A ) with radius ( AS ) intersects circle ( k ) at points ( E ) and ( F ). Since ( AS ) is the radius of ( k ), this circle is congruent to ( k ). - The intersection points ( E ) and ( F ) form equilateral triangles ( triangle AES ) and ( triangle AFS ), making ( angle SAF = 60^circ ).2. Coordinates and Vectors: - Place ( S ) at the origin ((0,0)), ( A ) at ((1,0)), and ( B ) at ((-1,0)). - Point ( D ) is determined by ( angle BAD = 28^circ ), leading to coordinates ( D ) as ((cos 124^circ, sin 124^circ)). - Point ( C ), diametrically opposite ( D ), has coordinates ((cos 56^circ, -sin 56^circ)).3. Intersection Points: - The circle centered at ( A ) intersects ( k ) at ( F ) with coordinates ((frac{1}{2}, frac{sqrt{3}}{2})).4. Angle Calculation: - Using vector analysis and dot product, the angle ( angle CFS ) is calculated by considering vectors ( overrightarrow{FC} ) and ( overrightarrow{FS} ). - The key identity used: ( 1 + sin 26^circ = 2 sin 58^circ cos 32^circ ), leading to ( cos theta = cos 32^circ ), hence ( theta = 32^circ ).Thus, the measure of ( angle CFS ) is (boxed{32^circ}).,<think>Okay, let me try to work through this geometry problem step by step. So, we have a circle k with center S, and diameters AB and CD. That means AB and CD are both straight lines passing through the center S, right? So, AB and CD intersect at S, the center. Also, the angle BAD is 28 degrees. Then there's another circle with center A that passes through S, so the radius of this circle is AS. This circle intersects the original circle k at points E and F, with D and F on the same side of line AB. We need to find the measure of angle CFS.First, let me try to visualize this. Let me sketch a rough diagram in my mind. There's a circle k, center S. AB and CD are diameters, so they cross at S. The angle BAD is 28 degrees. So point A is on the circle, and B is the other end of the diameter from A. Similarly, C and D are endpoints of another diameter. The angle at A between BA and DA is 28 degrees. So points B, A, D form an angle of 28 degrees at A.Now, there's another circle centered at A passing through S. Since S is the center of circle k, and the circle centered at A passes through S, the radius of this new circle is AS. Since AS is a radius of the original circle k, because S is the center. Wait, the original circle k has center S, so the radius of k is SA = SB = SC = SD. The circle centered at A with radius AS would then pass through S and intersect circle k at points E and F. So, E and F are points that lie on both circles k and the circle centered at A.Moreover, D and F are on the same side of line AB. So line AB is a diameter, so it divides the circle into two semicircles. D is on one side, and F is on the same side. So we need to figure out where E and F are located.Let me try to piece this together. Since AB and CD are diameters, their intersection at S means that AB and CD are perpendicular? Wait, no, not necessarily. Unless specified, diameters can be at any angle. But since angle BAD is 28 degrees, that might relate to their positions.Let me think. Points B, A, D with angle at A being 28 degrees. Since AB is a diameter, A and B are endpoints, so BA is a straight line through S. Similarly, CD is another diameter. So point D is somewhere on the circle such that angle BAD is 28 degrees. So if we imagine point A fixed, then point D is located such that when we go from B to A to D, the angle is 28 degrees. Since AB is a diameter, moving from B through S to A. So angle BAD is 28 degrees. Therefore, the arc from B to D around the circle would be related to this angle. But angle at A is 28 degrees, which is an inscribed angle. Wait, but A is on the circle, so angle BAD is an inscribed angle? Wait, actually, BA is a diameter, so BA is a straight line. So points B, A, and D are on the circle, with AB as a diameter. Then angle BAD is 28 degrees. So AD is a chord of the circle making a 28-degree angle with AB at point A.Since AB is a diameter, the angle subtended by AD at the center S would be twice that, so 56 degrees. Therefore, the arc BD would be 56 degrees. Wait, but angle BAD is 28 degrees, which is an inscribed angle. But since A is on the circle, angle BAD is actually a tangent angle? Wait, maybe not. Let's clarify.In a circle, the angle between two chords at a point on the circumference is equal to half the measure of the arc intercepted by the angle. So angle BAD is 28 degrees, so the arc BD that it intercepts would be 56 degrees. So the arc from B to D is 56 degrees. Therefore, the central angle BSD would be 56 degrees. So since S is the center, angles at S would correspond to the arcs. Therefore, angle BSD is 56 degrees. Therefore, the diameter CD is such that points C and D are endpoints, and angle BSD is 56 degrees.But wait, CD is another diameter. So CD passes through S and is a straight line. If angle BSD is 56 degrees, then since CD is a diameter, points C and D are separated by 180 degrees. Wait, but angle BSD is 56 degrees. So if S is the center, and B, S, D are points with angle at S between SB and SD being 56 degrees. But SB is a radius, as is SD. So triangle BSD is an isosceles triangle with SB = SD, and angle at S being 56 degrees. Therefore, the base angles at B and D would be (180 - 56)/2 = 62 degrees each. Hmm, not sure if that's necessary yet.Now, moving on to the circle centered at A passing through S. Since this circle has center A and radius AS, which is equal to the radius of the original circle k. Because in circle k, all radii are equal, so SA = SB = SC = SD. So the circle centered at A with radius AS will pass through S and intersect circle k at points E and F. So the intersection points of the two circles.The two circles: circle k with center S and radius r, and circle centered at A with radius AS = r. So the distance between centers S and A is equal to r (since SA is a radius of circle k). Therefore, the two circles (k and the one centered at A) intersect each other at two points E and F such that the distance between their centers is equal to the radius of each. This forms two equilateral triangles? Because if two circles have radii equal to the distance between their centers, the triangle formed by the centers and an intersection point is equilateral. So triangle SAS is equilateral? Wait, S is the center of k, and A is another center. Wait, the distance SA is equal to the radius of circle k, which is the same as the radius of the circle centered at A. Therefore, the intersection points E and F will form two equilateral triangles: SAE and SAF. Therefore, angles at A for both triangles would be 60 degrees.Wait, let me think again. If two circles have centers separated by a distance equal to their radius (which are equal), then the points of intersection form equilateral triangles. So yes, triangle SAS is a triangle with SA = AS = radius, but wait, SA is the same as AS. Wait, perhaps triangle SAE, where E is an intersection point. Since circle k has center S and radius SA, and circle A has center A and radius AS. So points E and F lie on both circles, so SE = SA (because E is on circle k) and AE = AS (because E is on circle A). Therefore, triangle SAE has SA = SE = AE, so it's equilateral. Therefore, angle at S is 60 degrees, angle at A is 60 degrees, and same at E. Therefore, the points E and F are located such that they form equilateral triangles with S and A.Therefore, the angle between SA and SE is 60 degrees. Since S is the center of circle k, and SA is a radius, moving 60 degrees from SA would give the position of E. Similarly, the other intersection point F would be on the other side.But the problem states that D and F are on the same side of line AB. So we need to figure out which of the intersection points is F. Since D is on one side of AB, F must be on that same side. So depending on the orientation, F is the one that's on the same side as D.So once we have F located, we need to find angle CFS. Point C is the other end of the diameter CD. So C is diametrically opposite D. So once we know where D is, C is on the opposite side.Given all that, let me try to reconstruct the positions.First, draw circle k with center S. Draw diameter AB horizontally, with A on the left, B on the right, S in the center. Then angle BAD is 28 degrees. So point D is somewhere above or below the line AB such that angle BAD is 28 degrees. Let me assume AB is horizontal, and angle BAD is above AB. So D is above AB. Then CD is a diameter, so C is directly opposite D, below AB.Now, the circle centered at A passing through S: since A is on the left end of AB, and the circle has radius AS (which is equal to SA = radius of k). So this circle will pass through S and intersect circle k at two points: one above AB and one below AB. But since D and F are on the same side of AB, and D is above AB, then F must be the intersection point above AB. So E would be the other intersection point below AB, but we are concerned with F.Therefore, point F is where the circle centered at A intersects circle k above AB. Since triangle SAF is equilateral, angle SAF is 60 degrees. Therefore, the angle between SA and AF is 60 degrees. Since SA is along AB towards S, then AF is making a 60-degree angle upwards from SA. Therefore, point F is 60 degrees above the line AS (which is the same as AB, since AB is a diameter through S).Wait, SA is from S to A, which is left along AB. If we are at center A, drawing a circle with radius AS, which is equal to SA. The intersection points E and F would be located such that they are 60 degrees from SA. Since triangle SAF is equilateral, angle at A is 60 degrees, so the arc from S to F on the circle centered at A is 60 degrees. But since S is the center of the original circle, and F is also on circle k, we need to relate their positions.Alternatively, considering both circles, the points E and F lie on both circles. So SE = SA (radius of k) and AE = AS (radius of circle at A). So triangle AES is equilateral, so angle EAS is 60 degrees.Therefore, in circle k, point F is located such that angle SAF is 60 degrees. Since SA is the radius from S to A, then moving 60 degrees from SA at center A would place F at 60 degrees. But since S is the center, and A is a point on the circle, this might correspond to a specific arc on circle k.Alternatively, maybe it's better to use coordinates to model this.Let me set up coordinate system. Let’s place center S at the origin (0,0). Let’s take AB as the horizontal diameter, so point A is (-r, 0), point B is (r, 0), where r is the radius of circle k. Then CD is another diameter, making an angle such that angle BAD is 28 degrees. Let's find coordinates of point D.Angle BAD is 28 degrees. Since point A is (-r,0), angle BAD is the angle between BA and DA. Vector BA goes from A to B, which is from (-r,0) to (r,0), so direction is along the positive x-axis. Vector DA goes from A to D. So angle between BA (positive x-axis) and DA is 28 degrees. Therefore, point D is located such that when viewed from A, the angle between AB (which is towards B) and AD is 28 degrees.But since D is on circle k, which is centered at S (0,0) with radius r. So coordinates of D can be determined based on this angle.Wait, perhaps using coordinates would complicate things, but maybe necessary.Let’s assume the radius of circle k is 1 for simplicity. So S is (0,0), A is (-1,0), B is (1,0). Then we need to find coordinates of D such that angle BAD is 28 degrees.Vector BA is from A to B: (1 - (-1), 0 - 0) = (2, 0).Vector DA is from A to D: (D_x - (-1), D_y - 0) = (D_x +1, D_y).The angle between BA and DA is 28 degrees. The formula for the angle between two vectors u and v is:cosθ = (u · v) / (|u||v|)So here, u = BA = (2, 0), v = DA = (D_x +1, D_y)Therefore,cos(28°) = [(2)(D_x +1) + (0)(D_y)] / (|BA| |DA|)|BA| = 2, |DA| = sqrt((D_x +1)^2 + D_y^2)But point D lies on circle k, so D_x^2 + D_y^2 = 1.So we have:cos(28°) = [2(D_x +1)] / [2 * sqrt((D_x +1)^2 + D_y^2)]Simplify numerator and denominator:cos(28°) = [2(D_x +1)] / [2 sqrt((D_x +1)^2 + D_y^2)] = (D_x +1) / sqrt((D_x +1)^2 + D_y^2)Square both sides:cos²(28°) = (D_x +1)^2 / [(D_x +1)^2 + D_y^2]Multiply both sides by denominator:cos²(28°)[(D_x +1)^2 + D_y^2] = (D_x +1)^2Bring right side to left:cos²(28°)[(D_x +1)^2 + D_y^2] - (D_x +1)^2 = 0Factor:(D_x +1)^2 [cos²(28°) - 1] + cos²(28°) D_y^2 = 0Note that cos²θ - 1 = -sin²θ, so:(D_x +1)^2 (-sin²28°) + cos²28° D_y^2 = 0Multiply both sides by -1:(D_x +1)^2 sin²28° - cos²28° D_y^2 = 0But since D is on the circle, D_x² + D_y² = 1. Let me express D_y² = 1 - D_x², substitute into the equation:(D_x +1)^2 sin²28° - cos²28° (1 - D_x²) = 0Expand (D_x +1)^2:(D_x² + 2D_x + 1) sin²28° - cos²28° (1 - D_x²) = 0Distribute:D_x² sin²28° + 2D_x sin²28° + sin²28° - cos²28° + D_x² cos²28° = 0Combine like terms:D_x² (sin²28° + cos²28°) + 2D_x sin²28° + sin²28° - cos²28° = 0Since sin²θ + cos²θ = 1:D_x² + 2D_x sin²28° + sin²28° - cos²28° = 0Note that sin²28° - cos²28° = -cos(56°) (using the identity cos2θ = cos²θ - sin²θ). So:D_x² + 2D_x sin²28° - cos56° = 0Let me compute the numerical values to simplify:sin²28° ≈ (0.4695)² ≈ 0.2198cos56° ≈ 0.5592So equation becomes:D_x² + 2 * 0.2198 * D_x - 0.5592 = 0Which is:D_x² + 0.4396 D_x - 0.5592 = 0Solving quadratic equation:D_x = [-0.4396 ± sqrt(0.4396² + 4 * 0.5592)] / 2Compute discriminant:0.4396² ≈ 0.19334 * 0.5592 ≈ 2.2368Total discriminant ≈ 0.1933 + 2.2368 ≈ 2.4301sqrt(2.4301) ≈ 1.559Therefore,D_x ≈ [-0.4396 ± 1.559]/2We have two solutions:D_x ≈ (-0.4396 + 1.559)/2 ≈ 1.1194/2 ≈ 0.5597D_x ≈ (-0.4396 - 1.559)/2 ≈ -1.9986/2 ≈ -0.9993But since D is on circle k (radius 1), D_x must be between -1 and 1. The second solution is approximately -0.9993, which is close to -1, which is point A. But point D cannot be point A because angle BAD would be undefined. Therefore, D must be the other solution: D_x ≈ 0.5597.Then D_y² = 1 - D_x² ≈ 1 - (0.5597)² ≈ 1 - 0.3133 ≈ 0.6867, so D_y ≈ ±0.8287. Since angle BAD is 28°, and we assumed D is above AB (so D_y positive), we take D_y ≈ 0.8287.Therefore, coordinates of D are approximately (0.5597, 0.8287). Then point C is diametrically opposite D, so C is (-0.5597, -0.8287).Now, circle centered at A (-1,0) passing through S (0,0). The radius is AS, which is the distance from A to S: sqrt((0 - (-1))² + (0 - 0)²) = 1. So circle centered at (-1,0) with radius 1. Equation: (x +1)^2 + y^2 = 1.Circle k is centered at S (0,0) with radius 1: x² + y² = 1.Find intersection points E and F. Solve the system:1. x² + y² = 12. (x +1)^2 + y^2 = 1Subtract equation 1 from equation 2:(x +1)^2 + y^2 - (x² + y²) = 1 - 1Expand:x² + 2x +1 + y^2 - x² - y^2 = 0Simplify:2x +1 = 0 => x = -0.5Therefore, the intersection points lie on x = -0.5. Substitute back into equation 1:(-0.5)^2 + y² = 1 => 0.25 + y² = 1 => y² = 0.75 => y = ±√(0.75) = ±(√3)/2 ≈ ±0.8660Therefore, points E and F are (-0.5, -√3/2) and (-0.5, √3/2). Since D is above AB (y positive), F is the one on the same side, so F is (-0.5, √3/2 ≈ 0.8660).Now, we need to find angle CFS. Points C, F, S.Point C is (-0.5597, -0.8287), F is (-0.5, √3/2 ≈ 0.8660), and S is (0,0).First, let's find the coordinates:C: (-0.5597, -0.8287)F: (-0.5, ≈0.8660)S: (0,0)We need to compute angle at F between points C, F, S. Wait, angle CFS is the angle at F between points C, F, S. Wait, no: angle CFS is the angle at point F between points C, F, S. Wait, no. The notation ∠CFS means the angle at point F between points C, F, S. Wait, actually, no. In geometry, ∠CFS is the angle at point F formed by points C, F, S. So vertex at F, with sides FC and FS.To compute this angle, we can use vector analysis. Compute vectors FC and FS, then find the angle between them.First, find coordinates of vectors:Vector FC = C - F = (-0.5597 - (-0.5), -0.8287 - 0.8660) ≈ (-0.0597, -1.6947)Vector FS = S - F = (0 - (-0.5), 0 - 0.8660) = (0.5, -0.8660)Now, the angle between vectors FC and FS can be found using the dot product formula:cosθ = (FC · FS) / (|FC| |FS|)Compute the dot product:FC · FS ≈ (-0.0597)(0.5) + (-1.6947)(-0.8660) ≈ (-0.02985) + (1.6947 * 0.8660)Calculate 1.6947 * 0.8660 ≈ 1.6947 * 0.866 ≈ 1.6947 * 0.8 + 1.6947 * 0.066 ≈ 1.3558 + 0.112 ≈ 1.4678Therefore, FC · FS ≈ -0.02985 + 1.4678 ≈ 1.43795Compute |FC|:|FC| ≈ sqrt((-0.0597)^2 + (-1.6947)^2) ≈ sqrt(0.00356 + 2.872) ≈ sqrt(2.8756) ≈ 1.696Compute |FS|:|FS| = sqrt(0.5^2 + (-0.8660)^2) = sqrt(0.25 + 0.75) = sqrt(1) = 1Therefore,cosθ ≈ 1.43795 / (1.696 * 1) ≈ 1.43795 / 1.696 ≈ 0.8476Therefore, θ ≈ arccos(0.8476) ≈ 32 degrees.Wait, arccos(0.8476) is approximately 32 degrees. Let me check:cos(30°) ≈ 0.866, cos(32°) ≈ 0.848. Yes, 0.848 is approximately 32 degrees. Therefore, angle CFS is approximately 32 degrees.But wait, the problem is likely expecting an exact value, not an approximate. Since in the problem, all given angles are in whole numbers (28 degrees), and the configuration involves equilateral triangles (60 degrees), perhaps the answer is 30 degrees? But according to the approximate calculation, it's 32 degrees.Hmm, there might be an exact solution here.Let me think again without coordinates. Since we have several key points:1. Triangle SAF is equilateral (since SA = AF = radius), so angle SAF = 60°, which makes angle FAS = 60°, so arc FS in circle k would be 60°, since it's the central angle. Wait, no, in circle k, S is the center, so the central angle for arc FS would be angle FSF, but point F is on the circle. Wait, no, the central angle corresponding to arc FS is angle F S S, which doesn't make sense. Wait, the central angle for arc FS is the angle at S between points F and S. Wait, no. If you have arc FS on circle k, the central angle is angle FSD, where D is another point? Wait, perhaps not. Let me clarify.In circle k, the central angle between two points is the angle at S between those two points. For example, arc BF corresponds to central angle BSF. Similarly, arc FS corresponds to angle FSS, which is zero. Wait, no. Wait, if we have two points F and C on the circle, the central angle is angle FSC. But in our case, to find relations, maybe we need to find arcs or use properties of cyclic quadrilaterals or something else.Alternatively, let's consider triangle CFS. Points C, F, S.We know coordinates approximately, but perhaps there is a better way. Let's recall that point F is the intersection of the two circles, forming an equilateral triangle with A and S. So triangle AFS is equilateral, so angle FAS = 60°, and FS = AS = 1 (radius).In circle k, point C is diametrically opposite D, so angle DSD' = 180°, but D is 56 degrees from B as we found earlier. Wait, earlier we had angle BAD = 28°, which corresponded to arc BD = 56°, so central angle BSD = 56°. Therefore, since CD is a diameter, angle DSC = 180°, so the position of C is 180° from D.Therefore, the central angle from B to D is 56°, and from D to C is 180°, so from B to C is 56° + 180° = 236°, but that might not be necessary.Alternatively, in circle k, point F is at (-0.5, √3/2), which is 60° above the negative x-axis. Wait, in coordinates, if S is at (0,0), then point F at (-0.5, √3/2) is located at 120° from the positive x-axis. Because the angle from positive x-axis to point F is 180° - 60° = 120°, since it's in the second quadrant.Similarly, point C is diametrically opposite D, which we found was at approximately (0.56, 0.83). So point D is at an angle of arctan(0.83/0.56) ≈ 56 degrees from the positive x-axis. Therefore, point C is at 56 + 180 = 236 degrees from positive x-axis, which is in the third quadrant.Therefore, point F is at 120°, point C is at 236°, and S is at the center (0,0). So angle CFS is the angle at F between points C, F, S.To find this angle, perhaps using the inscribed angle theorem or something related.Alternatively, since we have coordinates, let's compute the exact angle.Wait, point F is (-0.5, √3/2), which is (cos 120°, sin 120°). Because cos 120° = -0.5, sin 120° = √3/2. So F is at 120°, which makes sense since the circle centered at A passing through S intersects circle k at 60° from SA, which would place F at 120° from the positive x-axis.Point C is diametrically opposite D, which was at angle 28° + 180°? Wait, earlier we found coordinates for D as approximately (0.56, 0.83), which is angle arctan(0.83/0.56). Let me compute that: 0.83 / 0.56 ≈ 1.48, arctangent of 1.48 is about 56 degrees. So point D is at 56° from positive x-axis, so point C is at 56° + 180° = 236°.So point C is at 236°, point F is at 120°, and S is at 0°. We need the angle at F between C, F, S.Alternatively, since we know coordinates, we can calculate vectors:Vector FC = C - F = (cos236° - cos120°, sin236° - sin120°)Similarly, Vector FS = S - F = (-cos120°, -sin120°)But let's compute exact values.First, note that:cos120° = -1/2, sin120° = √3/2cos236° = cos(180° + 56°) = -cos56°, sin236° = -sin56°So:Vector FC = ( -cos56° - (-1/2), -sin56° - (√3/2) ) = ( -cos56° + 1/2, -sin56° - √3/2 )Vector FS = (0 - (-1/2), 0 - √3/2 ) = (1/2, -√3/2 )Now, compute the angle between these two vectors.Let’s denote FC as (x1, y1) and FS as (x2, y2).Compute the dot product:FC · FS = x1x2 + y1y2= [ (-cos56° + 1/2 ) * 1/2 ] + [ (-sin56° - √3/2 ) * (-√3/2 ) ]Simplify:First term: [ -cos56° * 1/2 + 1/2 * 1/2 ] = - (cos56°)/2 + 1/4Second term: [ sin56° * √3/2 + (√3/2 * √3/2 ) ] = (√3/2 sin56° ) + (3/4 )So total dot product:- (cos56°)/2 + 1/4 + (√3/2 sin56° ) + 3/4Combine constants: 1/4 + 3/4 = 1So:Dot product = 1 - (cos56°)/2 + (√3/2 sin56° )Factor out 1/2:= 1 + (1/2)( -cos56° + √3 sin56° )Now, note that -cos56° + √3 sin56° can be expressed as 2 sin(56° - 30° ) due to the formula:a sinθ + b cosθ = R sin(θ + φ), but here we have -cosθ + √3 sinθ.Let’s consider:√3 sin56° - cos56° = 2 sin(56° - 30° ) = 2 sin26°Because:The formula is: a sinθ - b cosθ = R sin(θ - φ), where R = √(a² + b²), φ = arctan(b/a)Here, a = √3, b = 1, so R = √( (√3)^2 + 1^2 ) = √4 = 2φ = arctan(1/√3) = 30°Therefore:√3 sin56° - cos56° = 2 sin(56° - 30° ) = 2 sin26°Therefore, the dot product becomes:1 + (1/2)(2 sin26° ) = 1 + sin26°Now, compute |FC| and |FS|:|FS| is the length of vector FS, which is sqrt( (1/2)^2 + (-√3/2)^2 ) = sqrt(1/4 + 3/4 ) = sqrt(1) = 1|FC| is the length of vector FC:sqrt( [ -cos56° + 1/2 ]^2 + [ -sin56° - √3/2 ]^2 )Let’s compute each component:First component squared:(-cos56° + 1/2)^2 = cos²56° - cos56° + 1/4Second component squared:(-sin56° - √3/2 )^2 = sin²56° + √3 sin56° + 3/4So |FC|² = cos²56° - cos56° + 1/4 + sin²56° + √3 sin56° + 3/4Combine terms:cos²56° + sin²56° = 1- cos56° + √3 sin56°1/4 + 3/4 = 1So |FC|² = 1 + 1 + (-cos56° + √3 sin56° ) = 2 + ( √3 sin56° - cos56° ) = 2 + 2 sin26° (from previous calculation)Therefore, |FC| = sqrt(2 + 2 sin26° ) = sqrt(2(1 + sin26° )) = sqrt(2) * sqrt(1 + sin26° )Now, returning to the dot product formula:cosθ = (1 + sin26° ) / ( sqrt(2) * sqrt(1 + sin26° ) * 1 )Simplify:cosθ = sqrt(1 + sin26° ) / sqrt(2 )Using the identity:sqrt( (1 + sinθ)/2 ) = cos(45° - θ/2 )Let’s verify:1 + sinθ = 2 cos²(45° - θ/2 )Let’s set φ = 45° - θ/2, then cos²φ = (1 + cos2φ)/2But 2φ = 90° - θ, so cos2φ = sinθ. Therefore,(1 + sinθ)/2 = cos²φ => sqrt( (1 + sinθ)/2 ) = |cosφ|Which is cosφ since phi is acute.Therefore,sqrt( (1 + sin26° ) / 2 ) = cos(45° - 13° ) = cos32°Therefore,cosθ = sqrt( (1 + sin26° ) / 2 ) = cos32°Therefore, θ = 32°, which matches the approximate calculation.But the problem states angle BAD is 28°, and we arrived at angle CFS = 32°. However, 32° is not a standard angle, unless there's a relation I missed.Wait, let's think differently. Maybe angle chasing.Given that SAF is equilateral, angle FAS = 60°. So arc FS in circle k is 60°, since angle at center S is equal to the arc measure. Therefore, arc FS = 60°, so central angle FSF is 60°? Wait, no. The central angle for arc FS is angle FSA, but S is the center. Wait, no. The central angle corresponding to arc FS is the angle at S between points F and S. Wait, that doesn't make sense because both are radii. Wait, the central angle for arc FC would be angle FSC, but in this case, point C is another point.Alternatively, since triangle SAF is equilateral, then angle at S is 60°, so angle ASF = 60°, which is the central angle for arc AF. Wait, arc AF would then be 60°, since central angles correspond to arcs.But in circle k, arc AF is 60°, so point F is 60° away from point A along the circumference. Since point A is at (-1,0), moving 60° counterclockwise would place F at angle 180° - 60° = 120°, which matches the coordinate calculation.Now, point C is diametrically opposite D, and angle BAD = 28°, so arc BD is 56°, which we calculated earlier. So point D is 56° away from B along the circumference. Therefore, point D is at 56° from B, so since B is at 0°, point D is at 56°, and point C is at 56° + 180° = 236°.Therefore, in circle k, point F is at 120°, point C is at 236°, and S is at center.To find angle CFS, which is the angle at F between points C, F, S. In circle geometry, this angle can be calculated using inscribed angles. The angle at F would be half the measure of the arc CS not containing F, or something like that. Wait, the inscribed angle theorem states that the measure of an angle formed by two chords in a circle is half the sum or difference of the arcs they intercept.Wait, more precisely, the measure of an angle formed by two chords intersecting at a point on the circumference is half the measure of the intercepted arc.But in this case, angle CFS is formed by points C, F, S. Wait, but S is the center. So it's an angle at F with one side connecting to C and the other to S. Since S is the center, FS is a radius. So perhaps we can consider triangle FSC.In triangle FSC, we know FS and SC are both radii of circle k, so FS = SC = 1 (if radius is 1). Wait, but SC is the distance from S to C, which is a radius, so yes, SC = 1. FS is also a radius, so FS = 1. Wait, but FC is the distance from F to C. If both FS and SC are 1, then triangle FSC is isosceles with FS = SC = 1.Wait, but earlier coordinates show that FC is not equal to FS or SC. Wait, in coordinate terms, distance from F (-0.5, √3/2) to C (-cos56°, -sin56°) is sqrt[ (-0.5 + cos56°)^2 + (√3/2 + sin56°)^2 ].But according to previous calculation, |FC| ≈ 1.696, which is not equal to 1. So triangle FSC is not isosceles. So maybe that approach was wrong.Alternatively, since S is the center, and we have points C and F on the circle, then angle CFS is related to the arcs. The angle at F between C and S would be half the measure of the arc CS that does not contain F.Wait, inscribed angle theorem: the measure of an angle formed by two points on the circumference is half the measure of the arc intercepted by those points. So angle at F intercepting arc CS. But since S is the center, CS is a diameter? No, CD is a diameter, but CS is not necessarily a diameter. Wait, CD is a diameter, so CS is a radius, same as SD. Hmm.Alternatively, the angle at F (angle CFS) intercepts arc CS. But since S is the center, the arc CS is actually the same as the central angle for arc CS, which is angle CSS, which is a straight line. Wait, no.Wait, maybe I need to think differently. Let's consider the arcs. The angle at F is angle CFS, which is formed by chords FC and FS. The inscribed angle theorem says that this angle is equal to half the measure of the arc intercepted by the angle, which would be arc CS. But wait, angle at F intercepts arc CS. But arc CS is from C to S, but S is the center. In the circle, all arcs are between two points on the circumference. But S is the center, not on the circumference. Therefore, this approach might not work.Wait, perhaps we need to use the law of cosines in triangle CFS.In triangle CFS, we can find the lengths of FC, FS, and CS, then apply the law of cosines to find angle at F.We have:FS = 1 (radius)CS = 1 (radius)FC: distance between C and F. Earlier calculated as sqrt(2 + 2 sin26° ) ≈ 1.696Wait, if FS = 1 and CS = 1, then triangle FSC has two sides equal to 1, and FC is the third side. Therefore, using the law of cosines:FC² = FS² + CS² - 2 * FS * CS * cos(angle FSC)But angle FSC is the central angle between FS and CS, which is the angle at S between points F, S, C. Which is the angle between points F and C as viewed from the center S.In circle k, point F is at 120°, point C is at 236°, so the central angle between them is 236° - 120° = 116°. Therefore, angle FSC = 116°.Therefore,FC² = 1² + 1² - 2 * 1 * 1 * cos116°Calculate:FC² = 2 - 2 cos116°But from previous calculation, FC² = 2 + 2 sin26°. Wait, this seems conflicting.Wait, let's check:Wait, 2 + 2 sin26° ≈ 2 + 2*0.4384 ≈ 2 + 0.8768 ≈ 2.8768From law of cosines:2 - 2 cos116° ≈ 2 - 2*(-0.5592) ≈ 2 + 1.1184 ≈ 3.1184Wait, these don't match. Hmm, so there must be an error.Wait, maybe the central angle is not 116°. Let me double-check. Point F is at 120°, point C is at 236°, so the angle between them is 236° - 120° = 116°, which is correct. But why the discrepancy in FC²?Wait, in the coordinate system, distance from F (-0.5, √3/2) to C (-cos56°, -sin56°):Let me use exact values.cos56° ≈ 0.5592, sin56° ≈ 0.8290F is (-0.5, √3/2 ≈ 0.8660)C is (-0.5592, -0.8290)Distance FC:sqrt[ (-0.5 + 0.5592)^2 + (0.8660 + 0.8290)^2 ]≈ sqrt[ (0.0592)^2 + (1.695)^2 ]≈ sqrt[ 0.0035 + 2.873 ]≈ sqrt[ 2.8765 ] ≈ 1.696Law of cosines gives FC² ≈ 2 - 2 cos116° ≈ 2 - 2*(-0.5592) ≈ 2 + 1.1184 ≈ 3.1184, which would mean FC ≈ sqrt(3.1184) ≈ 1.766, which contradicts the coordinate calculation.So there must be a mistake in the assumption.Wait, central angle FSC is 116°, so applying law of cosines:FC² = FS² + CS² - 2*FS*CS*cos(angle FSC)But FS and CS are both radii, length 1. So FC² = 1 + 1 - 2*1*1*cos(116°) ≈ 2 - 2*(-0.5592) ≈ 3.1184, but coordinate calculation gives FC ≈ 1.696, FC² ≈ 2.876.Therefore, inconsistency here implies an error in reasoning.Wait, wait, points F, S, C are not necessarily forming triangle FSC with FS and CS as sides. Because S is the center, FS and CS are radii, yes, but in triangle FSC, sides are FS, CS, and FC.Wait, but according to coordinates, distance from F to C is approximately 1.696, but law of cosines says it should be sqrt(2 - 2 cos116°). Let me compute cos116°:cos116° = cos(180° - 64°) = -cos64° ≈ -0.4384Therefore,FC² = 2 - 2*(-0.4384) = 2 + 0.8768 = 2.8768Therefore, FC ≈ sqrt(2.8768) ≈ 1.696, which matches the coordinate calculation. Therefore, my previous calculation was incorrect due to miscalculating cos116°. Cos116° is equal to -cos64° ≈ -0.4384, not -0.5592. Cos56° ≈ 0.5592, yes, but cos64° ≈ 0.4384.Therefore, FC ≈ sqrt(2.8768) ≈ 1.696, which matches. Therefore, angle at F can be found using law of cosines:In triangle FSC,cos(angle CFS) = (FC² + FS² - CS²) / (2*FC*FS)Wait, angle at F is angle CFS, so using law of cosines:cos(angle CFS) = (FC² + FS² - CS²) / (2*FC*FS)But FS = 1, FC ≈ 1.696, CS = 1.Therefore,cos(angle CFS) = ( (1.696)^2 + 1^2 - 1^2 ) / (2*1.696*1 )= (2.876 + 1 - 1) / 3.392= 2.876 / 3.392 ≈ 0.8476Which gives angle CFS ≈ 32°, same as before.But the problem is likely constructed so that angle CFS is 30°, given the equilateral triangles and 28°, but maybe not. Alternatively, perhaps the answer is 32°, but since the problem is in Chinese, maybe the exact value is 28° + 4° = 32°, or relates to other angles.Alternatively, maybe there's a geometric relation I'm missing.Let me consider triangle AFS, which is equilateral. So angle at A is 60°, angle at S is 60°, angle at F is 60°. But point F is also on circle k. Then, perhaps quadrilateral AFSC is something?Alternatively, consider that angle CFS is related to the other angles via cyclic quadrilaterals or something else.Alternatively, since angle BAD = 28°, and angle SAF = 60°, maybe we can find some relationship.Alternatively, construct line segments and find similar triangles.But given the time I've spent and the approximate calculation suggesting 32°, which is close to 30°, but not exact. However, in the exact calculation using the trigonometric identity, we found that angle CFS = 32°, since cos(theta) = cos32°, so theta = 32°.But is there an exact value? Let me see:We had:cos(angle CFS) = cos32°,therefore angle CFS = 32°, which is the exact value based on the previous steps.But where does 32° come from? It's 28° + 4°, but I don't see a direct relation. Alternatively, 90° - 58° = 32°, but 58° is 2*29°, not sure.Alternatively, 32° is half of 64°, which is the supplementary angle of 116°, the central angle.Wait, the central angle for arc FC is 116°, so the inscribed angle would be half of that, 58°, but angle CFS is not an inscribed angle for arc FC.Wait, angle CFS is at point F, intercepting arc CS. But S is the center, so the intercepted arc is from C to S, but S is the center, not on the circumference. Therefore, this approach is invalid.Alternatively, since FS and FC are chords of the circle, but angle at F is outside the central angle.Alternatively, using coordinates and exact values:We had:cos(angle CFS) = sqrt( (1 + sin26° ) / 2 )But sin26° is sin(56° - 30°), which is sin56° cos30° - cos56° sin30°, but not sure.Alternatively, express sin26° in terms of sin(angle BAD). Since angle BAD is 28°, which is close to 26°, but not directly.Alternatively, given that angle BAD is 28°, which corresponds to arc BD of 56°, and angle CFS is 32°, which is 56° - 24°, but again not directly.Given that the approximate calculation gives 32°, and the exact calculation using trigonometric identities also gives arccos(cos32°) = 32°, so the answer should be 32 degrees. Therefore, I think the answer is 32°, which should be written as boxed{32^circ}.But let me check once more for any possible geometric relations.Consider triangle AFS (equilateral) and triangle ABD (with angle at A = 28°). Maybe there is a relationship between these triangles.Point D is such that angle BAD = 28°, and point F is such that angle FAS = 60°. Perhaps quadrilateral ABFD has some properties.Alternatively, consider that angle SAF = 60°, angle BAD = 28°, so angle FAS = 60°, angle BAD = 28°, so angle FAD = 60° - 28° = 32°, which might correspond to angle FAD = 32°, but I don't see how this relates to angle CFS.Alternatively, in triangle AFD, angle at A is 32°, if F is 60° from A and D is 28° from B. But I need to visualize this.Alternatively, since angle CFS is found to be 32° via calculation, and there's no indication of a simpler exact angle, the answer is likely 32 degrees.Final Answerboxed{32^circ}</think>Given a circle ( k ) with center ( S ) and diameters ( overline{AB} ) and ( overline{CD} ), where ( angle BAD = 28^circ ). A circle centered at ( A ) passing through ( S ) intersects ( k ) at points ( E ) and ( F ), with ( D ) and ( F ) on the same side of line ( AB ). We need to determine ( angle CFS ).1. Identify Key Points and Angles: - ( overline{AB} ) and ( overline{CD} ) are diameters of circle ( k ), intersecting at center ( S ). - ( angle BAD = 28^circ ), implying the arc ( BD ) subtended by this angle is ( 56^circ ). - The circle centered at ( A ) passing through ( S ) (radius ( AS )) intersects ( k ) at ( E ) and ( F ).2. Equilateral Triangle Formation: - The circle centered at ( A ) with radius ( AS ) forms an equilateral triangle ( triangle SAF ) since ( SA = AF = AS ). - This implies ( angle SAF = 60^circ ).3. Coordinate System and Geometry: - Place ( S ) at the origin, ( A ) at ( (-1, 0) ), and ( B ) at ( (1, 0) ). - Point ( D ) is determined such that ( angle BAD = 28^circ ), and ( C ) is diametrically opposite ( D ).4. Intersection Points Calculation: - The circle centered at ( A ) intersects ( k ) at ( x = -0.5 ), giving points ( E ) and ( F ) with coordinates ( (-0.5, pm frac{sqrt{3}}{2}) ). - ( F ) is on the same side as ( D ), so ( F ) is ( (-0.5, frac{sqrt{3}}{2}) ).5. Vector Analysis and Dot Product: - Calculate vectors ( overrightarrow{FC} ) and ( overrightarrow{FS} ). - Use the dot product to find the angle between vectors, resulting in ( cos theta = 0.8476 ), leading to ( theta approx 32^circ ).6. Verification and Conclusion: - Using trigonometric identities and verifying with the law of cosines, the angle ( angle CFS ) is confirmed to be ( 32^circ ).Thus, the final answer is (boxed{32^circ}).]

🔑:## First solution.Triangle A F S is equilateral, so varangle A S F=60^{circ}. 1 pointTriangle C S F is isosceles, so varangle C F S=varangle S C F. 1 pointSince angles angle D C F and angle D A F are inscribed angles subtended by the chord overline{D F}, we havevarangle C F S=varangle D C F=varangle D A F .Finally, we concludevarangle D A F=varangle S A F-varangle B A D=60^{circ}-28^{circ}=32^{circ}

🧠:[<think>Okay, let's see. The problem is about two teams, A and B, working together to complete a project. Team A alone takes m days, and team B alone takes n days. We need to find out how many days they'll take if they work together. The options are given as A: 1, B: (m+n)/mn, C: mn/(m+n), D: m+n. Hmm.First, I remember that when dealing with work rates, it's usually about combining their rates. So, like, if one person can do a job in m days, their work rate is 1/m per day. Similarly, the other person would have a rate of 1/n per day. Then, working together, their combined rate would be the sum of their individual rates. Right?So let's break it down. Team A can finish the project in m days, so their work rate is 1/m projects per day. Similarly, team B's rate is 1/n projects per day. If they work together, their combined rate should be 1/m + 1/n. To add those fractions, I need a common denominator, which would be mn. So, converting them, that's n/(mn) + m/(mn) = (n + m)/mn. So together, they can complete (m + n)/mn of the project in one day.But the question is asking for the number of days needed to complete the entire project. Since their combined daily rate is (m + n)/mn, the time taken (let's call it T) would be the reciprocal of that rate. Because time is 1 divided by rate. So T = 1 / [(m + n)/mn] = mn/(m + n). That would be option C.Wait, let me double-check. Suppose m is 2 days and n is 2 days. Then working together, the formula gives 2*2/(2+2) = 4/4 = 1 day. Which makes sense, if both teams can finish in 2 days alone, together they'd take half the time? Wait, no. Wait, if one team takes 2 days alone, their rate is 1/2 per day. Two teams with the same rate would combine to 1/2 + 1/2 = 1 project per day, so time is 1 day. Which matches the calculation. So that example works.Another example: if team A takes 3 days and team B takes 6 days. Then combined rate is 1/3 + 1/6 = 1/2 per day. So time is 2 days. Using the formula, mn/(m + n) = 3*6/(3+6) = 18/9 = 2. Correct again.Alternatively, if m=1 and n=1, the formula gives 1/(1+1) = 1/2 day. But if both can finish in 1 day alone, working together they would finish in half a day. That makes sense. So the formula works here too.Looking at the answer options, option C is mn/(m + n), which is what we derived. So the correct answer should be C.Wait, just to check the other options. Option B is (m + n)/mn, which is the combined rate per day, but not the time. So if someone confuses rate with time, they might pick B, but actually, the time is reciprocal. Option D is m + n, which would be if they added the times, but that's incorrect because when working together, the time should be less than either m or n. Unless one of them is slower, but even then, it's not additive. Option A is 1, which is a trick if m and n are 1, but generally not. So yes, C is correct.Final Answerboxed{C}</think>To determine the number of days required for team A and team B to complete the project together, we start by analyzing their individual work rates. Team A's work rate is (frac{1}{m}) projects per day, and team B's work rate is (frac{1}{n}) projects per day. When working together, their combined work rate is the sum of their individual rates:[frac{1}{m} + frac{1}{n}]To add these fractions, we find a common denominator, which is (mn):[frac{n}{mn} + frac{m}{mn} = frac{m + n}{mn}]This combined rate (frac{m + n}{mn}) represents the fraction of the project completed in one day. To find the number of days required to complete the entire project, we take the reciprocal of this combined rate:[frac{1}{frac{m + n}{mn}} = frac{mn}{m + n}]Thus, the number of days required for team A and team B to complete the project together is (frac{mn}{m + n}), which corresponds to option C.[boxed{C}],<think>Okay, so I need to figure out how many days it takes for team A and team B to complete a project together, right? Let me think. The problem says team A alone takes m days, and team B alone takes n days. The answer choices are given, so maybe I can work through it step by step.First, I remember that when dealing with work rates, it's usually about how much each can do in a day. So if team A can finish the project in m days, then their work rate is 1/m per day. Similarly, team B's work rate is 1/n per day. Yeah, that makes sense because if they take m days, each day they do 1/m of the project.So when they work together, their rates should add up, right? So combined, they can do 1/m + 1/n per day. Let me write that down: combined work rate = (1/m) + (1/n). To add those two fractions, I need a common denominator. The common denominator of m and n is mn. So converting 1/m to n/(mn) and 1/n to m/(mn). Then adding them together: (n + m)/mn. So the combined work rate is (m + n)/mn per day.But the question is asking for the number of days needed to complete the project. Since work rate is reciprocal of time, the time taken should be 1 divided by the combined work rate. So the total time T is 1 / ((m + n)/mn) = mn/(m + n). Let me check the answer choices. Option C is mn/(m + n), which matches what I got. But wait, let me verify to make sure I didn't mix anything up.Another way to think about it: suppose team A can do the job in 10 days, so m = 10. Team B can do it in 15 days, n = 15. Then combined rate is 1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6. So time taken is 6 days. Using the formula mn/(m + n) = (10*15)/(10+15) = 150/25 = 6. Perfect, that works. So if m=10 and n=15, answer is 6, which is indeed 150/25. So the formula holds here.Alternatively, if both teams take the same time, say m=5 and n=5. Then combined rate is 2/5, so time is 5/2 = 2.5 days. Using the formula mn/(m+n) = 25/10 = 2.5. That also checks out. So seems like the formula is correct. Therefore, the answer should be option C.Final Answerboxed{C}</think>To determine the number of days required for team A and team B to complete the project together, we start by calculating their individual work rates. Team A's work rate is (frac{1}{m}) per day, and team B's work rate is (frac{1}{n}) per day. When working together, their combined work rate is the sum of their individual work rates:[frac{1}{m} + frac{1}{n}]To add these fractions, we find a common denominator:[frac{1}{m} + frac{1}{n} = frac{n}{mn} + frac{m}{mn} = frac{m + n}{mn}]The combined work rate is (frac{m + n}{mn}) per day. The number of days required to complete the project is the reciprocal of this combined work rate:[frac{1}{frac{m + n}{mn}} = frac{mn}{m + n}]Thus, the required number of days when team A and team B work together is (frac{mn}{m + n}), which corresponds to option C.[boxed{C}]]

🔑:Let the total work be 1. The work rates of team A and B are respectively frac{1}{m} and frac{1}{n}. According to the problem, we have 1 div left( frac{1}{m} + frac{1}{n} right) = frac{mn}{m+n}. Therefore, the correct choice is boxed{text{C}}.

🧠:[<think>Okay, so I need to find three distinct natural numbers such that their sum is 55, and the sum of any two of them is a perfect square. Let me think through this step by step.First, let's denote the three numbers as a, b, and c, where a, b, c are distinct natural numbers. The problem states that a + b + c = 55, and also that a + b, a + c, and b + c are all perfect squares. Let me write down these equations:1. a + b + c = 552. a + b = S₁² (some perfect square)3. a + c = S₂²4. b + c = S₃²Since a, b, c are natural numbers, S₁², S₂², S₃² must also be positive integers, and since the numbers are distinct, the squares must also be distinct. Also, each pair sum must be a different perfect square.Now, if I add all three pair sums together: (a + b) + (a + c) + (b + c) = 2(a + b + c) = 2*55 = 110. So S₁² + S₂² + S₃² = 110.Therefore, the three perfect squares must add up to 110. So my task reduces to finding three distinct perfect squares S₁², S₂², S₃² such that their sum is 110, and also that each of them is less than 55 (since each pair sum is less than the total sum 55). Wait, no, actually, the pair sum a + b can be up to 54 (since the numbers are distinct natural numbers; the maximum a + b would be if c is 1, so a + b = 54). But 54 isn't a perfect square. So maybe the maximum possible perfect square less than 55 is 49 (7²). Wait, but 49 is 7², which is 49. So S₁², S₂², S₃² are all less than 55, but adding up to 110. So let's see.So first, I need to find three distinct perfect squares less than 55 that add up to 110. Let's list all perfect squares less than 55:1² = 12² = 43² = 94² = 165² = 256² = 367² = 498² = 64 (but 64 is greater than 55, so stop here)So the possible squares are 1, 4, 9, 16, 25, 36, 49.Now, we need three distinct squares from this list that add up to 110. Let's think about combinations.First, note that 49 is the largest square here. Let's see if 49 is part of the combination. Then the remaining two squares need to add up to 110 - 49 = 61. Let's see if there are two distinct squares in our list that add up to 61.Looking for pairs:36 + 25 = 61. Yes, 36 and 25. So 49, 36, 25. Let's check if these are distinct: 25, 36, 49 are all distinct. So that's one possible combination.Another possible pair for 61: 49 + 12? 12 isn't a square. 16 + 45? 45 isn't a square. So only 36 + 25 works. So that gives us the triplet 25, 36, 49.Another combination: maybe 36 is the largest square. Then 110 - 36 = 74. Now, 74 needs to be split into two squares. Let's see: 49 + 25 = 74. So 36, 49, 25. But that's the same as before, just ordered differently.Alternatively, 64 is too big, so next is 49. If we don't include 49, maybe 36 is the next. Then 110 - 36 = 74. But as before, 49 + 25 = 74, which includes 49, so we still need 49.If we try 25 as the largest square: 110 -25 =85. Then looking for two squares adding to 85. 81 is 9², but 81 +4=85. 81 isn't in our list (since 9²=81, but 81 is greater than 55? Wait no, 85 would be the sum of two squares. But 81 is 9², which is 81, which is greater than 55. So 85 can be 49 + 36 = 85. So again, 25, 49, 36. Same triplet.Alternatively, check if there are other triplets. Let's see:If we try 16 as one of the squares: 16 + something + something else =110. Then the other two squares need to add to 94. Let's see if 49 +45=94. 45 isn't a square. 36 + 58=94. 58 isn't. 25 + 69=94. 69 isn't. So no. So seems like 25, 36, 49 is the only triplet.Alternatively, check 9, 16, ... Hmm, 9 +16 + something=110. 110-25=85. Again, 85 as before. So same issue.Alternatively, maybe 1,4, but that's too small. So maybe the only triplet is 25, 36, 49. Let me check another possibility. For example, 16 + 25 + 49 = 90, which is less than 110. Wait, no. Wait, 16 +25 +49=90, which is not 110. So no.Wait, 25 + 36 +49=110. Yes, that's correct. So the three squares must be 25, 36, 49. So S₁²=25, S₂²=36, S₃²=49, or any permutation. But since the numbers a, b, c are distinct, we need to assign these squares appropriately.So let's denote:Let me assume that:a + b = 25a + c = 36b + c = 49But these assignments might vary. Let's check if these equations can be solved for a, b, c.From these three equations:We can add the first two: (a + b) + (a + c) = 25 + 36 = 61Which is 2a + b + c = 61But since b + c =49, we can substitute:2a +49=61 → 2a=12 → a=6Then from a + b =25, b=25 -6=19From a + c=36, c=36 -6=30Check if b + c=19 +30=49. Yes, that works.So the numbers would be a=6, b=19, c=30. Let's check if they are distinct natural numbers: 6,19,30 are distinct and natural. Their sum is 6+19+30=55, which matches the total. Each pair sum:6+19=25=5²6+30=36=6²19+30=49=7²Perfect, all perfect squares. So this seems to satisfy all conditions.But wait, is this the only possible combination? Let me check another permutation of the squares. Suppose we assign:a + b =25a + c=49b + c=36Then solving:From a + b=25a + c=49Subtracting: c - b=24From b + c=36So we have:c = b +24Substitute into b + c=36:b + (b +24)=36 → 2b +24=36 → 2b=12 →b=6Then c=6 +24=30Then a=25 -b=25 -6=19So numbers are a=19, b=6, c=30. Same as before, just reordered. So the three numbers are 6,19,30 in some order.Another permutation: Assign a + b=36, a + c=25, b + c=49.Let's try:a + b=36a + c=25So c=25 -aFrom b + c=49: b + (25 -a)=49 → b=49 -25 +a=24 +aBut from a + b=36: a + (24 +a)=36 → 2a +24=36 →2a=12→a=6Then b=24 +6=30c=25 -6=19Again, same numbers. So regardless of the permutation of the squares, we end up with the same triplet. So the three numbers must be 6, 19, 30.But let's check if there's another triplet of squares that add to 110. Let's see. For example, 16 + 36 + 49=101, which is less than 110. 25 + 25 + 60, but duplicates aren't allowed. 16 + 25 + 49=90, not 110. 1 + 4 + 105=110, but 105 isn't a square. Maybe 9 + 16 + 85=110, but 85 isn't a square. 4 + 16 + 90=110, nope. 9 + 36 + 65=110, 65 isn't a square. 16 + 25 + 69=110, 69 isn't a square. So seems like 25, 36, 49 is the only combination. Therefore, the three numbers must be 6, 19, 30. Let me double-check:6 +19 +30=55. Correct.6+19=25=5²6+30=36=6²19+30=49=7²All perfect squares, as required. And all numbers are distinct natural numbers. So the answer is 6, 19, and 30.Final AnswerThe three natural numbers are boxed{6}, boxed{19}, and boxed{30}.</think>Given three distinct natural numbers (a), (b), and (c) such that their sum is 55, and the sum of any two of these numbers is a perfect square, we need to find these numbers.We denote the equations as follows:1. (a + b + c = 55)2. (a + b = S_1^2)3. (a + c = S_2^2)4. (b + c = S_3^2)Adding these three equations, we get:[(a + b) + (a + c) + (b + c) = S_1^2 + S_2^2 + S_3^2 = 2(a + b + c) = 2 times 55 = 110]Thus, we need three distinct perfect squares that add up to 110. The possible perfect squares less than 55 are (1, 4, 9, 16, 25, 36, 49). We find that the triplet (25, 36, 49) adds up to 110.We solve the system of equations for (a), (b), and (c):1. (a + b = 25)2. (a + c = 36)3. (b + c = 49)Adding the first two equations:[(a + b) + (a + c) = 25 + 36 implies 2a + b + c = 61]Substituting (b + c = 49) into the equation:[2a + 49 = 61 implies 2a = 12 implies a = 6]Substituting (a = 6) into (a + b = 25):[6 + b = 25 implies b = 19]Substituting (a = 6) into (a + c = 36):[6 + c = 36 implies c = 30]Verifying the sums:- (a + b = 6 + 19 = 25 = 5^2)- (a + c = 6 + 30 = 36 = 6^2)- (b + c = 19 + 30 = 49 = 7^2)All conditions are satisfied, so the three natural numbers are (boxed{6}), (boxed{19}), and (boxed{30}).,<think>Okay, let's see. I need to find three distinct natural numbers such that their sum is 55, and the sum of any two of them is a perfect square. Hmm, that sounds tricky but maybe I can break it down step by step.First, let me denote the three numbers as a, b, and c, all distinct natural numbers. So, according to the problem:a + b + c = 55And also, the sum of any two should be a perfect square:a + b = S1²a + c = S2²b + c = S3²Where S1, S2, S3 are some integers. Since all numbers are natural, these squares must be at least 1, but since the numbers are distinct and positive, the smallest possible sum of two would be 1 + 2 = 3, so the squares need to be at least 3²? Wait, but 3² is 9. Wait, no, if a and b are natural numbers, their sum could be as small as 1 + 2 = 3, but 3 isn't a perfect square. So maybe the minimal possible square here is 4 (2²). But 1 + 3 = 4, so maybe the numbers are at least 1, 3, and something else. But since all three numbers need to be distinct, and their total sum is 55, they can't be too small.Let me think. Let's suppose that a, b, c are positive integers, distinct, and each pair sums to a square. Let's denote the three equations:1. a + b = S1²2. a + c = S2²3. b + c = S3²We also know that a + b + c = 55. If I add all three equations together, I get:(a + b) + (a + c) + (b + c) = S1² + S2² + S3²Which simplifies to:2(a + b + c) = S1² + S2² + S3²But since a + b + c = 55, then:2*55 = S1² + S2² + S3² => 110 = S1² + S2² + S3²So the sum of the three squares must be 110. Now, I need to find three squares that add up to 110, and each of these squares corresponds to the sum of two distinct natural numbers. Also, each of these squares must be such that when you subtract the numbers from them, you get the third number. Let me elaborate.If a + b = S1², then c = 55 - S1²Similarly, a + c = S2² => b = 55 - S2²And b + c = S3² => a = 55 - S3²So each number is 55 minus one of the squares. Therefore, each square must be less than 55 because a, b, c are positive. So S1², S2², S3² are all less than 55. So S1, S2, S3 are integers such that their squares are less than 55. So possible squares are 1, 4, 9, 16, 25, 36, 49. So possible square values are up to 49 since 64 is 8² which is 64 >55.So possible squares: 1,4,9,16,25,36,49. But since a, b, c are natural numbers, their sums must be at least 3 (1+2). So the squares must be at least 4 (since 3 is not a square). So possible squares are 4,9,16,25,36,49.Therefore, S1, S2, S3 can be integers from 2 up to 7, since 7²=49, and 8²=64 is too big.So the possible squares are 4,9,16,25,36,49.Now, the sum of these three squares must be 110. So we need three squares from that list (4,9,16,25,36,49) that add up to 110. Let's check combinations.First, let me list all possible triples of these squares and see which ones add up to 110.Start with the largest square, 49.If one of the squares is 49, then the remaining two squares need to add up to 110 - 49 = 61.Now, can two squares from the list (4,9,16,25,36) add up to 61?Check 36 + 25 = 61. Yes, that works. So 49, 36, 25.Another combination: 49 + 16 + (something). 49 + 16 = 65, so remaining would be 45, which isn't a square. 49 + 9 = 58, remaining 52, not a square. 49 + 4 = 53, remaining 57, not a square. So only 49 +36 +25=110.Another possibility: if the largest square is 36. Then remaining sum is 110 -36=74. Need two squares adding to 74. Check 49 +25=74. So 36,49,25. But that's same as previous. Or 36 + 36 + ? But duplicates? Wait, but the numbers need to be distinct. Wait, the three squares can be same? Wait, in the problem statement, the three numbers a,b,c are distinct, but the squares could potentially repeat? Wait, let me check. If two different pairs sum to the same square, that's allowed? Let me see. Suppose, for example, a + b = 25, and a + c =25. Then, since a + b = a + c, that would imply b = c, but the problem states that the three numbers must be distinct. Therefore, the squares must also be distinct. Because if two pairs have the same sum, then two variables would be equal, which is not allowed. So S1², S2², S3² must all be distinct. Therefore, the three squares must be distinct.Therefore, in the previous case, 49,36,25 are all distinct and sum to 110, so that's valid.Another possible combination: Let's see, if the largest square is 25. Then remaining sum is 110 -25=85. Check if two squares add up to 85. 81 +4=85, but 81 isn't in our list. 64 not in list. 49 +36=85. So 49,36,25 again. So same as before.Alternatively, 16 + 81=97, no. So seems like the only combination is 49,36,25.Another check: What if all three squares are 36, 36, 38. But 38 isn't a square. So that's invalid. Or 25,25,60. Nope. So the only possible triplet is 49,36,25.Wait, let me check other possibilities. Suppose we don't take 49. Let's say the largest square is 36. Then the sum of the other two squares must be 74. As before, 49 +25=74, but that requires 49 which is larger than 36. Wait, but if we don't use 49, then 36 is the largest. Let's see: 36 + 25 + 49 is the same as before. If we exclude 49, then can we get 36 + 25 + something? 36 +25=61, 110 -61=49, which is again 49. So no other combinations.Alternatively, let's check all possible triplets:Possible squares: 4,9,16,25,36,49.Looking for three distinct squares summing to 110.Let's list all possible triplets:1. 49,36,25: 49+36+25=110 ✔️2. 49,36,16: 49+36+16=101 ❌3. 49,36,9: 49+36+9=94 ❌4. 49,36,4: 49+36+4=89 ❌5. 49,25,16: 49+25+16=90 ❌6. 49,25,9: 49+25+9=83 ❌7. 49,25,4: 49+25+4=78 ❌8. 49,16,9: 49+16+9=74 ❌9. 49,16,4: 49+16+4=69 ❌10. 49,9,4: 49+9+4=62 ❌Now with 36 as the next largest:11. 36,25,16: 36+25+16=77 ❌12. 36,25,9: 36+25+9=70 ❌13. 36,25,4: 36+25+4=65 ❌14. 36,16,9: 36+16+9=61 ❌15. 36,16,4: 36+16+4=56 ❌16. 36,9,4: 36+9+4=49 ❌Continuing with 25:17. 25,16,9: 25+16+9=50 ❌18. Etc. Clearly none of the smaller triplets will reach 110.So the only triplet of distinct squares that sum to 110 is 49, 36, 25. Therefore, S1², S2², S3² must be 49, 36, and 25 in some order.Now, let's assign these squares to the pairs:a + b = 49a + c = 36b + c = 25Wait, but let's check if this assignment is possible. Let's solve these equations.If a + b = 49a + c = 36b + c = 25Then, subtract the second equation from the first: (a + b) - (a + c) = 49 -36 => b - c =13From the third equation: b + c =25So we have:b - c =13b + c =25Adding these two equations: 2b =38 => b=19Then c=25 -b=6Then a=49 -b=30So a=30, b=19, c=6Check if they are distinct natural numbers: 30,19,6 are distinct and positive. Their sum: 30+19+6=55 ✔️Check the pairwise sums:30 +19=49=7² ✔️30 +6=36=6² ✔️19 +6=25=5² ✔️Perfect! So that works.Wait, but let's check if there could be other assignments. For example, suppose we permute the squares:Case 1: a + b =25, a + c=36, b +c=49Let's solve:From a + b =25a + c=36Subtract: c - b=11From b + c=49So c - b=11 and c + b=49Adding: 2c=60 => c=30, then b=49 -30=19, a=25 -19=6So we get a=6, b=19, c=30. Same numbers, just reordered. So the numbers are 6,19,30.Case 2: a + b=25, a +c=49, b +c=36Solve:a + b=25a +c=49Subtract: c - b=24From b +c=36So c - b=24 and c + b=36Adding: 2c=60 => c=30, then b=36 -30=6, a=25 -6=19So a=19, b=6, c=30. Again same numbers.Similarly, other permutations will just give permutations of the same three numbers. Therefore, the three numbers are 6,19,30 in some order.Therefore, the answer should be these three numbers. Let me verify once more.Sum: 6 +19 +30=55 ✔️Pairwise sums:6+19=25=5² ✔️6+30=36=6² ✔️19+30=49=7² ✔️All perfect squares. All numbers are distinct natural numbers. Perfect.So the three natural numbers are 6, 19, and 30.Final AnswerThe three natural numbers are boxed{6}, boxed{19}, and boxed{30}.</think>Given three distinct natural numbers (a), (b), and (c) such that their sum is 55 and the sum of any two of these numbers is a perfect square, we need to find these numbers.We start with the equations:[a + b + c = 55]and the sums of pairs being perfect squares:[a + b = S_1^2, quad a + c = S_2^2, quad b + c = S_3^2]Adding these three equations, we get:[(a + b) + (a + c) + (b + c) = S_1^2 + S_2^2 + S_3^2]which simplifies to:[2(a + b + c) = S_1^2 + S_2^2 + S_3^2]Given (a + b + c = 55), we substitute to find:[2 times 55 = S_1^2 + S_2^2 + S_3^2 implies 110 = S_1^2 + S_2^2 + S_3^2]We need three distinct perfect squares that sum to 110. The possible squares (less than 55) are 4, 9, 16, 25, 36, and 49. The only combination of three distinct squares that sum to 110 is 49, 36, and 25.We solve the system of equations for these squares:[a + b = 49, quad a + c = 36, quad b + c = 25]Subtracting the second equation from the first:[(a + b) - (a + c) = 49 - 36 implies b - c = 13]From the third equation:[b + c = 25]Adding these two results:[(b - c) + (b + c) = 13 + 25 implies 2b = 38 implies b = 19]Substituting (b = 19) into (b + c = 25):[19 + c = 25 implies c = 6]Substituting (b = 19) into (a + b = 49):[a + 19 = 49 implies a = 30]Thus, the numbers are (a = 30), (b = 19), and (c = 6). Verifying the sums:[30 + 19 + 6 = 55][30 + 19 = 49, quad 30 + 6 = 36, quad 19 + 6 = 25]All sums are perfect squares.The three natural numbers are boxed{6}, boxed{19}, and boxed{30}.]

🔑:【Analysis and Solution】Three natural numbers are all different, so the three sums of any two of them, which are perfect squares, are also different;Adding the three perfect squares is equivalent to twice the sum of these three natural numbers;Therefore, decompose 55 times 2=110 into the sum of three perfect squares;The perfect squares not exceeding 110 are the following 11: 0^{2}=0, 1^{2}=1, 2^{2}=4, 3^{2}=9, 4^{2}=16, 5^{2}=25, 6^{2}=36, 7^{2}=49, 8^{2}=64, 9^{2}=81, 10^{2}=100; 110 div 3=36 frac{2}{3}, so there must be a perfect square greater than 36 frac{2}{3};Since the sum of the three different natural numbers is 55, these three perfect squares do not exceed 55; hence, one of the perfect squares must be 7^{2}=49;110=49+61=49+36+25, that is, these three perfect squares are 49, 36, 25; 55-49=6, 55-36=19, 55-25=30, that is, these three natural numbers are 6, 19, 30.

🧠:[<think>Okay, so I need to find the third term in the expansion of (1 - x)(1 + 2x)^5 when it's arranged in ascending powers of x. Hmm, let's see. I remember that when expanding expressions like this, the binomial theorem is useful. Let me recall the binomial theorem. The expansion of (a + b)^n is the sum from k=0 to n of (n choose k) * a^(n - k) * b^k. Right, so I can use that to expand (1 + 2x)^5 first, and then multiply the result by (1 - x). After that, I can combine like terms and arrange them in ascending powers of x. The third term would then be the coefficient of x^2, since the first term is the constant term (x^0), the second is x^1, and the third is x^2. Wait, is that correct? Let me make sure. If arranged in ascending powers, the first term is the constant term, then x, then x^2, so yes, the third term is the x^2 term. So my goal is to find the coefficient of x^2 in the expansion. Let me work through this step by step.First, expand (1 + 2x)^5 using the binomial theorem. Let's do that. The expansion would be the sum from k=0 to 5 of C(5, k) * 1^(5 - k) * (2x)^k. Simplifying each term, that would be C(5, k) * (2x)^k. So each term is C(5, k) * 2^k * x^k. So let's compute each term:For k=0: C(5,0) * 2^0 * x^0 = 1 * 1 * 1 = 1k=1: C(5,1) * 2^1 * x^1 = 5 * 2 * x = 10xk=2: C(5,2) * 2^2 * x^2 = 10 * 4 * x^2 = 40x^2k=3: C(5,3) * 2^3 * x^3 = 10 * 8 * x^3 = 80x^3k=4: C(5,4) * 2^4 * x^4 = 5 * 16 * x^4 = 80x^4k=5: C(5,5) * 2^5 * x^5 = 1 * 32 * x^5 = 32x^5So putting it all together, the expansion of (1 + 2x)^5 is:1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5Now, we need to multiply this by (1 - x). Let's write that out:(1 - x)(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5)To multiply this out, we distribute each term in (1 - x) across all the terms in the expansion. So:1*(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) + (-x)*(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5)Let's compute each part separately.First, multiplying by 1:1*1 = 11*10x = 10x1*40x^2 = 40x^21*80x^3 = 80x^31*80x^4 = 80x^41*32x^5 = 32x^5Then, multiplying by (-x):(-x)*1 = -x(-x)*10x = -10x^2(-x)*40x^2 = -40x^3(-x)*80x^3 = -80x^4(-x)*80x^4 = -80x^5(-x)*32x^5 = -32x^6Now, let's combine these two results.First, write out all the terms from the first multiplication:1, 10x, 40x^2, 80x^3, 80x^4, 32x^5Then the terms from the second multiplication:- x, -10x^2, -40x^3, -80x^4, -80x^5, -32x^6Now, combine like terms. Let's arrange them by ascending powers of x.Constant term (x^0): 1x term: 10x - x = 9xx^2 term: 40x^2 -10x^2 = 30x^2x^3 term: 80x^3 -40x^3 = 40x^3x^4 term: 80x^4 -80x^4 = 0x^4 (which cancels out)x^5 term: 32x^5 -80x^5 = -48x^5x^6 term: -32x^6So putting it all together, the expanded form is:1 + 9x + 30x^2 + 40x^3 -48x^5 -32x^6Wait a minute, the x^4 term cancels out? Let me check that again. From the first multiplication, we have 80x^4, and from the second multiplication, -80x^4. Yes, 80x^4 -80x^4 = 0. So there is no x^4 term. Interesting.But let's verify if that's correct.Original expansion of (1 + 2x)^5 is up to x^5. Then multiplying by (1 - x) would indeed combine terms. Let's recheck the coefficients:For x^0: Only 1*1 = 1For x^1: 1*10x + (-x)*1 = 10x - x = 9xFor x^2: 1*40x^2 + (-x)*10x = 40x^2 -10x^2 = 30x^2For x^3: 1*80x^3 + (-x)*40x^2 = 80x^3 -40x^3 = 40x^3Wait, hold on. Wait, no. Wait, when multiplying by -x, the term that gives x^3 is (-x)*40x^2 = -40x^3. But in the original expansion, the coefficient of x^3 is 80x^3 from the first multiplication. So 80x^3 -40x^3 = 40x^3. Correct.Similarly, for x^4: From first multiplication, 80x^4, and from second multiplication, (-x)*80x^3 = -80x^4. So 80x^4 -80x^4 = 0. Correct.x^5: From first multiplication, 32x^5, and from second multiplication, (-x)*80x^4 = -80x^5. So 32x^5 -80x^5 = -48x^5. Correct.x^6: Only from second multiplication, (-x)*32x^5 = -32x^6. Correct.So the combined expansion is indeed 1 + 9x + 30x^2 + 40x^3 -48x^5 -32x^6. Therefore, arranged in ascending powers of x, the terms are:1 (x^0),9x (x^1),30x^2 (x^2),40x^3 (x^3),-48x^5 (x^5),-32x^6 (x^6)Wait, but there's no x^4 term, as it canceled out. So in ascending order, the terms go from x^0 to x^6, with x^4 missing. Therefore, the third term in the expansion is 30x^2. Hence, the third term is 30x^2. Therefore, the answer should be 30x^2, and the problem asks for the third term, which would be 30x^2. But let me double-check if "third term" refers to the coefficient or the term itself. The problem says "Determine the third term in the expansion... when arranged in ascending powers of x." So the third term is 30x^2. If they want just the coefficient, it's 30, but the problem says "the third term", which would be the term itself, which is 30x². However, sometimes in math problems, "term" can refer to the coefficient when the variable is implied, but given the way the question is phrased, it's safer to check. Wait, the original problem says "Determine the third term in the expansion of the expression (1 - x)(1 + 2x)^5 when arranged in ascending powers of x." So when arranged in ascending powers, the order is 1, then 9x, then 30x², so the third term is 30x². Therefore, if they want the term, it's 30x², but maybe they just want the coefficient. Wait, the problem might be expecting the answer in boxed notation, as per the initial instruction. The user said "put your final answer within boxed{}". So in math problems like this, usually, unless specified otherwise, the term is given as the coefficient. Wait, but no, sometimes the term is written as the entire term. Let me check standard terminology. For example, if someone asks for the third term in the expansion of (a + b)^n, arranged in ascending powers, they usually expect the term itself, like "the term 30x²" or written as boxed{30x^2}. But let me check the original problem again. It says "Determine the third term in the expansion...". So in the expansion, once arranged in ascending powers, the third term is 30x², so the answer is 30x², which would be boxed as boxed{30x^2}. Alternatively, maybe they just want the coefficient, but given that the problem says "third term", I think it's more likely the term itself, which is 30x².But to confirm, let's consider the expansion:1 + 9x + 30x² + 40x³ -48x⁵ -32x⁶So the terms are:1st term: 12nd term: 9x3rd term: 30x²4th term: 40x³5th term: -48x⁵6th term: -32x⁶Therefore, the third term is indeed 30x². So the answer should be boxed{30x^2}Wait, but maybe the question is from an exam where they expect just the coefficient? Hmm. Let me check similar problems. For example, if you have an expansion and they ask for a specific term, often in math exams, especially in the context of binomial expansions, they sometimes refer to the term as the coefficient. But in this case, the problem says "the third term in the expansion", and expansions are usually written as a sum of terms, each being a coefficient times x to some power. So the third term would be the entire 30x². However, sometimes people refer to "terms" as just the coefficients. Wait, but the term is 30x². So unless specified otherwise, the term is 30x², which includes both the coefficient and the variable part. So I think the correct answer is 30x², so boxed as boxed{30x^2}However, to be absolutely sure, let's see if multiplying (1 - x)(1 + 2x)^5 could have been approached differently, maybe by considering only up to the x² term, since we need the third term, which is x². Because sometimes, if you only need a specific term, you can compute the necessary coefficients without expanding the entire expression. Let me try that approach as a verification.We need the coefficient of x² in the expansion of (1 - x)(1 + 2x)^5. Let's denote (1 + 2x)^5 as A. Then, (1 - x) * A = A - xA. Therefore, the coefficient of x² in the product is equal to (coefficient of x² in A) minus (coefficient of x¹ in A). So, if I can find the coefficient of x² in A and subtract the coefficient of x¹ in A, that would give the coefficient of x² in the final expansion.Given that A = (1 + 2x)^5, the coefficient of x^k in A is C(5, k) * 2^k.Therefore, coefficient of x² in A is C(5, 2)*2² = 10*4 = 40Coefficient of x¹ in A is C(5, 1)*2¹ = 5*2 = 10Therefore, coefficient of x² in (1 - x)A is 40 - 10 = 30. So the coefficient of x² is 30, which means the term is 30x². Therefore, this method confirms that the third term is 30x².Thus, regardless of the method used, the third term is 30x². So the answer is boxed{30x^2}Final AnswerThe third term in the expansion is boxed{30x^2}.</think>To determine the third term in the expansion of the expression ((1 - x)(1 + 2x)^5) when arranged in ascending powers of (x), we can use the binomial theorem to expand ((1 + 2x)^5) first and then multiply the result by ((1 - x)).1. Expanding ((1 + 2x)^5) using the binomial theorem: [ (1 + 2x)^5 = sum_{k=0}^{5} binom{5}{k} (2x)^k ] Calculating each term: [ begin{aligned} &k=0: binom{5}{0} (2x)^0 = 1, &k=1: binom{5}{1} (2x)^1 = 10x, &k=2: binom{5}{2} (2x)^2 = 40x^2, &k=3: binom{5}{3} (2x)^3 = 80x^3, &k=4: binom{5}{4} (2x)^4 = 80x^4, &k=5: binom{5}{5} (2x)^5 = 32x^5. end{aligned} ] Thus, the expansion is: [ 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5 ]2. Multiplying by ((1 - x)): [ (1 - x)(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) ] Distributing each term: [ begin{aligned} &1 cdot (1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) &- x cdot (1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) end{aligned} ] Combining like terms: [ begin{aligned} &1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5 &- x - 10x^2 - 40x^3 - 80x^4 - 80x^5 - 32x^6 end{aligned} ] Simplifying: [ 1 + 9x + 30x^2 + 40x^3 - 48x^5 - 32x^6 ]3. Identifying the third term: The expansion arranged in ascending powers of (x) is: [ 1 + 9x + 30x^2 + 40x^3 - 48x^5 - 32x^6 ] The third term is (30x^2).Thus, the third term in the expansion is (boxed{30x^2}).,<think>Okay, so I need to find the third term in the expansion of (1 - x)(1 + 2x)^5 when it's arranged in ascending powers of x. Hmm, ascending powers of x means starting from the constant term (x^0) and going up to higher powers. So the third term would be the coefficient of x^2, right? Because the first term is x^0, second is x^1, third is x^2. Yeah, that makes sense.First, I remember that when expanding expressions like this, it's helpful to use the binomial theorem for the (1 + 2x)^5 part and then multiply each term by (1 - x). So maybe I should expand (1 + 2x)^5 first, up to the x^2 term, since we only need up to x^2 for the third term. Wait, actually, if we multiply by (1 - x), the terms might interact. Let me think.The expansion of (1 + 2x)^5 can be done using the binomial formula. The binomial theorem says that (a + b)^n = sum from k=0 to n of (n choose k) a^(n - k) b^k. Here, a is 1 and b is 2x. So each term will be (5 choose k) * 1^(5 - k) * (2x)^k. That simplifies to (5 choose k) * 2^k * x^k.So let me compute the first three terms of (1 + 2x)^5:- When k = 0: (5 choose 0) * 2^0 * x^0 = 1 * 1 * 1 = 1- When k = 1: (5 choose 1) * 2^1 * x^1 = 5 * 2 * x = 10x- When k = 2: (5 choose 2) * 2^2 * x^2 = 10 * 4 * x^2 = 40x^2So the expansion up to x^2 is 1 + 10x + 40x^2 + ... higher terms. But since we need to multiply this by (1 - x), maybe we don't need to compute higher terms because multiplying by (1 - x) will mix the degrees. Let me check.Multiplying (1 - x) by (1 + 10x + 40x^2 + ...) would be:1*(1 + 10x + 40x^2) - x*(1 + 10x + 40x^2)So that's (1 + 10x + 40x^2) - (x + 10x^2 + 40x^3)Combine like terms:Constant term: 1x term: 10x - x = 9xx^2 term: 40x^2 - 10x^2 = 30x^2x^3 term: -40x^3, but since we need only up to the third term (x^2), maybe the x^3 term is not needed. Wait, but when arranged in ascending powers, the third term is x^2. So the coefficient there is 30. Therefore, the third term is 30x^2. So the answer is 30x^2, so the coefficient is 30. But wait, let me verify if I did that correctly.Alternatively, maybe I should do a more thorough expansion. Let's see. Let's expand (1 + 2x)^5 completely first, then multiply by (1 - x). But expanding the entire thing might be time-consuming, but since we need only up to x^2 terms, maybe the way I did it before is okay.Wait, but let's check again. Let me compute (1 + 2x)^5 using binomial coefficients. The coefficients for (1 + 2x)^5 are:Term 0: C(5,0)*1^5*(2x)^0 = 1*1*1 = 1Term 1: C(5,1)*1^4*(2x)^1 = 5*1*2x = 10xTerm 2: C(5,2)*1^3*(2x)^2 = 10*1*4x^2 = 40x^2Term 3: C(5,3)*1^2*(2x)^3 = 10*1*8x^3 = 80x^3Term 4: C(5,4)*1^1*(2x)^4 = 5*1*16x^4 = 80x^4Term 5: C(5,5)*1^0*(2x)^5 = 1*1*32x^5 = 32x^5So (1 + 2x)^5 = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5.Now multiply this by (1 - x):(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) * (1 - x)Multiply each term in the first polynomial by each term in the second polynomial:1*(1) + 1*(-x) + 10x*(1) + 10x*(-x) + 40x^2*(1) + 40x^2*(-x) + ... and so on.But since we only need up to x^2 terms, let's compute those:1*1 = 11*(-x) = -x10x*1 = 10x10x*(-x) = -10x^240x^2*1 = 40x^240x^2*(-x) = -40x^3 (but we can ignore x^3 and higher)So adding up the terms up to x^2:Constant term: 1x terms: -x + 10x = 9xx^2 terms: -10x^2 + 40x^2 = 30x^2Therefore, the expansion is 1 + 9x + 30x^2 + ... higher order terms.Therefore, the third term in ascending powers of x is 30x^2, so the coefficient is 30.Wait, that seems straightforward, but let me check if I made an error in multiplying terms. For example:1*(1) = 1Then, the next term is 1*(-x) + 10x*(1) = -x + 10x = 9xThen, the x^2 terms are 10x*(-x) + 40x^2*(1) = -10x^2 + 40x^2 = 30x^2Yes, that's correct. So the coefficient of x^2 is 30, so the third term is 30x^2.Alternatively, maybe I can use another approach. The third term in the expansion is the coefficient of x^2. So, another way is to think of the product (1 - x)(1 + 2x)^5. To find the coefficient of x^2, we can use the binomial coefficients.Let me recall that when multiplying two polynomials, the coefficient of x^k is the sum over i + j = k of the coefficients of x^i in the first polynomial multiplied by the coefficients of x^j in the second polynomial.Here, the first polynomial is (1 - x), which can be written as 1 + (-1)x + 0x^2 + ... So coefficients are a0 = 1, a1 = -1, and the rest are 0.The second polynomial is (1 + 2x)^5. Let me denote its coefficients as b0, b1, b2, ..., b5.Then, the coefficient of x^2 in the product will be a0*b2 + a1*b1 + a2*b0. But since a2 = 0, it's just a0*b2 + a1*b1.So compute that:a0 = 1, a1 = -1b1 is the coefficient of x^1 in (1 + 2x)^5, which we found earlier as 10.b2 is the coefficient of x^2 in (1 + 2x)^5, which is 40.Therefore, coefficient of x^2 in the product is 1*40 + (-1)*10 = 40 - 10 = 30.Therefore, the third term is 30x^2. So the answer is 30. That's the same result as before.Therefore, I think the third term is 30x^2, so the answer should be boxed{30}.Final AnswerThe third term in the expansion is boxed{30}.</think>To determine the third term in the expansion of ((1 - x)(1 + 2x)^5) when arranged in ascending powers of (x), we need to find the coefficient of (x^2).First, we expand ((1 + 2x)^5) using the binomial theorem:[(1 + 2x)^5 = sum_{k=0}^{5} binom{5}{k} (2x)^k]The first few terms of this expansion are:[(1 + 2x)^5 = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5]Next, we multiply this expansion by ((1 - x)):[(1 - x)(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5)]We focus on the terms up to (x^2):- The constant term: (1 cdot 1 = 1)- The (x) term: (1 cdot 10x - x cdot 1 = 10x - x = 9x)- The (x^2) term: (1 cdot 40x^2 - x cdot 10x = 40x^2 - 10x^2 = 30x^2)Thus, the coefficient of (x^2) in the expansion is 30. Therefore, the third term in the expansion is (boxed{30}).]

🔑:The general term of the expansion of (1+2x)^{5} is given by T_{r+1}=C_{5}^{r}×(2x)^{5-r}. Thus, the x^{2} term in the expansion of (1-x)(1+2x)^{5} is 30x^{2}.To find the third term, we need to determine the coefficient of x^{2}. Expanding (1+2x)^{5} using the binomial theorem gives:(1+2x)^{5} = C_{5}^{0}(2x)^{5} + C_{5}^{1}(2x)^{4} + C_{5}^{2}(2x)^{3} + C_{5}^{3}(2x)^{2} + C_{5}^{4}(2x)^{1} + C_{5}^{5}(2x)^{0}Next, we multiply each term by (1-x):begin{align}(1-x)(1+2x)^{5} &= C_{5}^{0}(2x)^{5} - C_{5}^{0}(2x)^{6} + C_{5}^{1}(2x)^{4} - C_{5}^{1}(2x)^{5} + C_{5}^{2}(2x)^{3} - C_{5}^{2}(2x)^{4} &+ C_{5}^{3}(2x)^{2} - C_{5}^{3}(2x)^{3} + C_{5}^{4}(2x)^{1} - C_{5}^{4}(2x)^{2} + C_{5}^{5}(2x)^{0} - C_{5}^{5}(2x)^{1} end{align}The x^{2} term is C_{5}^{3}(2x)^{2} - C_{5}^{4}(2x)^{2} = 30x^{2}.Therefore, the third term is boxed{30x^{2}}.

🧠:[<think>Okay, so I have this problem where I need to analyze the function F(x) = ln x + a/x, where a is a positive real number. The problem has two parts: first, finding the intervals where F(x) is increasing or decreasing, and second, determining the minimum value of a such that the slope of the tangent line to F(x) on the interval (0, 3] doesn't exceed 1/2. Let me tackle each part step by step.Starting with part (1): Monotonicity of F(x). To find where a function is increasing or decreasing, I remember that we need to look at its first derivative. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing. So first, let's compute F'(x).Given F(x) = ln x + a/x. The derivative of ln x is 1/x, and the derivative of a/x is -a/x². So putting that together:F'(x) = d/dx [ln x] + d/dx [a/x] = 1/x - a/x².Hmm, so F'(x) = (1/x) - (a/x²). Let me write that more neatly:F'(x) = (1/x) - (a/x²) = (x - a)/x².Wait, is that correct? Let me check:1/x is equivalent to x^(-1), derivative is -1 * x^(-2) = -1/x². Wait, no. Wait, no: the derivative of ln x is 1/x, correct. Then derivative of a/x: a is a constant, so derivative of a/x is a * derivative of x^(-1) = a * (-1)x^(-2) = -a/x². So yes, F'(x) = 1/x - a/x². Then combining those terms over a common denominator:Multiply numerator and denominator to get (x - a)/x². Yes, that's right. So F'(x) = (x - a)/x².So, critical points occur where F'(x) = 0 or where F'(x) is undefined. F'(x) is a rational function, undefined at x = 0. But the domain of F(x) is x > 0 because of the ln x term. So x is in (0, ∞). Therefore, the critical points are where (x - a)/x² = 0. That happens when x - a = 0, so x = a.Therefore, the critical point is at x = a. Now, to determine intervals of increase and decrease, we can test intervals around x = a. Since the denominator x² is always positive for x > 0, the sign of F'(x) depends on the numerator, x - a.So, when x > a, then x - a > 0, so F'(x) > 0, meaning the function is increasing. When x < a, x - a < 0, so F'(x) < 0, meaning the function is decreasing. Therefore, F(x) is decreasing on (0, a) and increasing on (a, ∞). So that's part (1) done. The intervals of monotonicity are decreasing on (0, a) and increasing on (a, ∞).Now, moving on to part (2). We need to find the minimum value of a such that the slope k of the tangent line at any point on the graph of y = F(x) for x in (0, 3] satisfies k ≤ 1/2. So, the slope k is the derivative F'(x), right? So F'(x) = (x - a)/x², and we need this to be ≤ 1/2 for all x in (0, 3].But wait, actually, the problem states that the slope k of the tangent line at any point on the graph of y = F(x) (for x in (0, 3]) always satisfies k ≤ 1/2. So, we need to ensure that F'(x) ≤ 1/2 for all x in (0, 3]. Therefore, we need to find the minimum value of a > 0 such that (x - a)/x² ≤ 1/2 for all x in (0, 3].So, the inequality is (x - a)/x² ≤ 1/2. Let's solve for a. Multiply both sides by x². But wait, since x is in (0, 3], x² is positive, so multiplying both sides by x² preserves the inequality direction. So:x - a ≤ (1/2)x²Then, rearranging:-a ≤ (1/2)x² - xMultiply both sides by -1, which reverses the inequality:a ≥ x - (1/2)x²So, a ≥ x - (1/2)x² for all x in (0, 3]. Therefore, to satisfy this inequality for all x in (0, 3], a must be greater than or equal to the maximum value of x - (1/2)x² on the interval (0, 3]. Therefore, the minimum value of a is equal to the maximum of x - (1/2)x² on (0, 3].So now, we need to find the maximum value of the function h(x) = x - (1/2)x² on the interval (0, 3]. Let's compute this. To find the maximum of h(x), take its derivative:h'(x) = 1 - xSet derivative equal to zero: 1 - x = 0 => x = 1.So, critical point at x = 1. Now, we can test the value of h(x) at x = 1 and at the endpoints (though the interval is (0, 3], so approaching x=0 from the right and at x=3.Compute h(1): 1 - (1/2)(1)^2 = 1 - 0.5 = 0.5Compute h(3): 3 - (1/2)(9) = 3 - 4.5 = -1.5As x approaches 0 from the right, h(x) = x - (1/2)x² approaches 0 - 0 = 0.Therefore, the maximum of h(x) on (0, 3] is 0.5 at x = 1. Therefore, the minimum value of a is 0.5. Therefore, a must be at least 0.5 to satisfy the inequality for all x in (0, 3].Wait, but let me verify this. So, if a is 0.5, then the inequality is x - 0.5 ≤ (1/2)x², so rearranged:x - 0.5 ≤ (1/2)x²Multiply both sides by 2:2x - 1 ≤ x²Bring all terms to left:x² - 2x + 1 ≥ 0Which factors as (x - 1)^2 ≥ 0, which is always true. So equality holds when x = 1. So when a = 0.5, the inequality (x - a)/x² ≤ 1/2 becomes equality at x = 1, and for other x, it's strictly less. Therefore, the maximum value of h(x) is 0.5, so setting a = 0.5 ensures that for all x in (0, 3], F'(x) ≤ 1/2. Therefore, the minimal a is 0.5.But wait, let me check if this is correct. Let's take a specific example. Let a = 0.5. Then F'(x) = (x - 0.5)/x². At x = 1, F'(1) = (1 - 0.5)/1 = 0.5, which is equal to 1/2. At x approaching 0+, F'(x) = (0 - 0.5)/x² = -0.5/x², which tends to negative infinity. But the problem states that the slope k must always satisfy k ≤ 1/2. But when x approaches 0, the slope becomes very negative, which is certainly less than 1/2. At x = 3, F'(3) = (3 - 0.5)/9 = 2.5/9 ≈ 0.277..., which is less than 0.5. So, the maximum slope occurs at x = 1, which is exactly 0.5. So, setting a = 0.5 ensures that the maximum slope is 1/2, and all other slopes are less than that. Therefore, this seems correct.But wait, let's make sure there isn't a higher value of h(x) somewhere else in the interval. Wait, we already found that h(x) = x - 0.5x² has its maximum at x =1 with h(1) = 0.5, and at x approaching 0, h(x) approaches 0, and at x=3, h(3) = -1.5, which is lower. So yes, the maximum is indeed 0.5, so a must be at least 0.5.But let me test with a smaller a. Suppose a = 0.4. Then h(x) = x - 0.4, and the inequality would be (x - 0.4)/x² ≤ 1/2. Let's test at x =1. (1 - 0.4)/1 = 0.6 ≤ 0.5? No, 0.6 is greater than 0.5. So that would violate the condition. Therefore, a cannot be less than 0.5. Hence, 0.5 is indeed the minimal value.Therefore, the answer to part (2) is a = 0.5.But wait, let me confirm once more. The key idea is that the maximum of (x - a)/x² over x in (0, 3] must be ≤ 1/2. Therefore, the maximum value of (x - a)/x² is equal to the maximum of h(x) = (x - a)/x². Wait, no, in the previous steps, we rearranged the inequality (x - a)/x² ≤ 1/2 to a ≥ x - (1/2)x². Therefore, the maximum of x - (1/2)x² is 0.5, so a must be at least 0.5.Alternatively, if we consider the original derivative F'(x) = (x - a)/x². We need this to be ≤ 1/2 for all x in (0, 3]. Let's treat this as an inequality:(x - a)/x² ≤ 1/2Multiply both sides by x² (positive, so inequality direction remains):x - a ≤ (1/2)x²Rearranged:a ≥ x - (1/2)x²So, a must be greater than or equal to the maximum value of x - (1/2)x² over (0, 3]. As we found, the maximum is 0.5 at x=1. Therefore, a ≥ 0.5. The minimal a is 0.5.So, yes, seems correct.But wait another way to think about this is to find the maximum of F'(x) on (0, 3], set that maximum equal to 1/2, then solve for a. So F'(x) = (x - a)/x². Let's find its maximum on (0, 3].To find maximum of (x - a)/x², take derivative with respect to x:Let’s set g(x) = (x - a)/x². Find its critical points.g'(x) = [1 * x² - (x - a) * 2x] / x^4 = [x² - 2x(x - a)] / x^4 = [x² - 2x² + 2a x] / x^4 = (-x² + 2a x)/x^4 = (-x + 2a)/x³.Set derivative equal to zero:(-x + 2a)/x³ = 0 => -x + 2a = 0 => x = 2a.So critical point at x = 2a. Now, the maximum of g(x) = (x - a)/x² occurs at x = 2a. So, evaluate g(x) at x = 2a:g(2a) = (2a - a)/( (2a)^2 ) = a/(4a²) = 1/(4a).Therefore, the maximum value of F'(x) is 1/(4a). Wait, but hold on. If the critical point is at x = 2a, but we need to check whether x = 2a is within our interval (0, 3]. So, if 2a ≤ 3, then the maximum is at x = 2a, otherwise, the maximum is at x=3. Wait, but actually, when we take the derivative of g(x), we found that the critical point is at x=2a, but we have to check whether this critical point is within the interval (0, 3].So, if 2a ≤ 3, i.e., a ≤ 3/2, then the maximum of g(x) is at x=2a, which is 1/(4a). If a > 3/2, then the critical point x=2a is outside the interval (0, 3], so the maximum would be at the endpoint x=3. Wait, but let's think about this.Wait, no. Because when a varies, the critical point x=2a might be inside or outside the interval (0, 3]. So, depending on a, the maximum could be at x=2a (if x=2a ≤ 3) or at x=3. Wait, but also, when x approaches 0, g(x) = (x - a)/x² approaches -a/x², which tends to negative infinity. So the maximum of g(x) on (0, 3] occurs either at x=2a (if 2a is in the interval) or at x=3.But hold on, let's actually do this properly. Suppose we want to find the maximum of F'(x) = (x - a)/x² on (0, 3].First, compute critical points by setting derivative of F'(x) to zero, which we did above, leading to x=2a. So, if x=2a is in (0, 3], then the critical point is inside the interval. If 2a > 3, then the critical point is outside, and we check endpoints.So, two cases:Case 1: 2a ≤ 3 => a ≤ 1.5. Then, maximum of F'(x) is at x=2a, which is 1/(4a).Case 2: 2a > 3 => a > 1.5. Then, the maximum of F'(x) on (0, 3] is at x=3. Let's compute F'(3):F'(3) = (3 - a)/9.So, in this case, the maximum is (3 - a)/9.But wait, but since we are supposed to have F'(x) ≤ 1/2 for all x in (0, 3], then depending on a, the maximum of F'(x) is either 1/(4a) (if a ≤ 1.5) or (3 - a)/9 (if a > 1.5). So we need both of these to be ≤ 1/2.So:If a ≤ 1.5, then 1/(4a) ≤ 1/2. Solving this inequality:1/(4a) ≤ 1/2Multiply both sides by 4a (since a > 0):1 ≤ 2a=> a ≥ 0.5But in this case, a ≤ 1.5. So combining, 0.5 ≤ a ≤ 1.5.If a > 1.5, then the maximum is at x=3: (3 - a)/9 ≤ 1/2Solving:(3 - a)/9 ≤ 1/2Multiply both sides by 9:3 - a ≤ 4.5=> -a ≤ 1.5=> a ≥ -1.5But since a > 1.5, this condition is automatically satisfied. Wait, but (3 - a)/9 is negative when a > 3. So, if a > 3, then (3 - a)/9 is negative, so the maximum of F'(x) would actually be at x approaching 0, but that tends to negative infinity. Wait, but the problem states that the slope k must satisfy k ≤ 1/2. Since if a > 3, then at x=3, the slope is negative, and all other slopes are negative or even more negative. So, in that case, all slopes are ≤ 1/2, which is true. But if a is between 1.5 and 3, then (3 - a)/9 is positive when a < 3. For example, if a = 2, then (3 - 2)/9 = 1/9 ≈ 0.111 < 0.5. So, in that case, the maximum slope is 1/(4a) when a ≤ 1.5, but when a > 1.5, the maximum slope is at x=3, but since a >1.5, (3 - a)/9 is less than (3 - 1.5)/9 = 1.5/9 = 1/6 ≈ 0.1667 < 0.5. So, actually, when a >1.5, the maximum slope is less than 1/6, which is way below 1/2. Therefore, for a >1.5, the slope condition is automatically satisfied. However, for a ≤1.5, the maximum slope is 1/(4a), which needs to be ≤1/2, leading to a ≥0.5. Therefore, the minimal a is 0.5.Therefore, the minimal value of a is 0.5. So, part (2)'s answer is 0.5.But let me check with a=0.5. Then, the maximum slope is at x=2a=1, which is 1/(4*0.5)=1/2. So exactly 1/2, which satisfies the condition k ≤1/2. So, that's correct. If a is less than 0.5, say a=0.4, then the maximum slope is 1/(4*0.4)=0.625, which is greater than 1/2, violating the condition. So, a must be at least 0.5.Therefore, the minimal a is 0.5.Hence, the answers are:(1) F(x) is decreasing on (0, a) and increasing on (a, ∞).(2) The minimum value of a is 0.5.But since the problem asks for the minimum value of the real number a in part (2), the answer is 0.5. Expressed as a fraction, that's 1/2. So, boxed, it would be boxed{dfrac{1}{2}}.Final AnswerThe minimum value of the real number ( a ) is boxed{dfrac{1}{2}}.</think>To solve the given problem, we start by analyzing the function ( F(x) = ln x + frac{a}{x} ) where ( a > 0 ).(1) Finding the intervals of monotonicity for ( F(x) ):First, we compute the first derivative of ( F(x) ):[F'(x) = frac{1}{x} - frac{a}{x^2} = frac{x - a}{x^2}]The critical point occurs where ( F'(x) = 0 ), which is at ( x = a ). Since the denominator ( x^2 ) is always positive for ( x > 0 ), the sign of ( F'(x) ) depends on the numerator ( x - a ):- For ( x < a ), ( x - a < 0 ), so ( F'(x) < 0 ) and ( F(x) ) is decreasing.- For ( x > a ), ( x - a > 0 ), so ( F'(x) > 0 ) and ( F(x) ) is increasing.Thus, ( F(x) ) is decreasing on ( (0, a) ) and increasing on ( (a, infty) ).(2) Finding the minimum value of ( a ) such that the slope of the tangent line at any point on ( y = F(x) ) for ( x in (0, 3] ) is always ( leq frac{1}{2} ):We need ( F'(x) leq frac{1}{2} ) for all ( x in (0, 3] ). This translates to:[frac{x - a}{x^2} leq frac{1}{2}]Rearranging the inequality:[x - a leq frac{1}{2}x^2 implies a geq x - frac{1}{2}x^2]We need to find the maximum value of ( h(x) = x - frac{1}{2}x^2 ) on ( (0, 3] ). Taking the derivative of ( h(x) ):[h'(x) = 1 - x]Setting ( h'(x) = 0 ) gives ( x = 1 ). Evaluating ( h(x) ) at critical points and endpoints:- ( h(1) = 1 - frac{1}{2}(1)^2 = frac{1}{2} )- ( h(3) = 3 - frac{1}{2}(9) = -1.5 )- As ( x to 0^+ ), ( h(x) to 0 )The maximum value of ( h(x) ) on ( (0, 3] ) is ( frac{1}{2} ). Therefore, the minimum value of ( a ) is ( frac{1}{2} ).Thus, the minimum value of ( a ) is boxed{dfrac{1}{2}}.,<think>Okay, so I need to solve this problem about the function F(x) which is the sum of f(x) = ln x and g(x) = a/x where a is a positive real number. The problem has two parts. The first part is to find the intervals where F(x) is increasing or decreasing. The second part is a bit more complex; it asks for the minimum value of a such that the slope of the tangent line (which is the derivative of F(x)) at any point on the interval (0, 3] is less than or equal to 1/2. Let me tackle each part step by step.Starting with part (1): Finding the intervals of monotonicity for F(x). Monotonicity refers to where the function is increasing or decreasing, and that's determined by the first derivative. So, I need to compute F'(x) and analyze its sign.Given F(x) = ln x + a/x. To find F'(x), I'll differentiate each term separately.The derivative of ln x is 1/x. The derivative of a/x can be written as a * x^(-1), so using the power rule, that's -a * x^(-2) or -a/x². Therefore, putting it together:F'(x) = 1/x - a/x².Now, to find where F'(x) is positive or negative, I need to analyze this expression. Let's write F'(x) as (1/x - a/x²). Let's combine these terms over a common denominator:F'(x) = (x - a)/x².So, F'(x) = (x - a)/x². Since x is in the domain of F(x), which is (0, ∞) because ln x is only defined for x > 0 and a/x is also defined for x > 0 (since a > 0).Now, to determine where F'(x) is positive or negative, note that the denominator x² is always positive for x > 0. Therefore, the sign of F'(x) is determined by the numerator: x - a.So, when x - a > 0, that is, when x > a, F'(x) is positive, meaning F is increasing.When x - a < 0, which is when x < a, F'(x) is negative, meaning F is decreasing.Therefore, F(x) is decreasing on (0, a) and increasing on (a, ∞).But wait, the problem didn't specify a domain for x here in part (1), so I think we can state the intervals as (0, a) decreasing and (a, ∞) increasing. However, since a is a positive real number, the critical point is at x = a. So, the function has a minimum at x = a because it decreases before and increases after.But the question just asks for intervals of monotonicity, not the extrema. So, part (1)'s answer should be:F(x) is decreasing on (0, a) and increasing on (a, ∞).But let me verify my steps again. The derivative is (x - a)/x². Since x² is always positive, the sign is indeed determined by x - a. So, yes, when x > a, derivative positive, increasing; when x < a, derivative negative, decreasing.So part (1) is done. Now moving on to part (2). The problem states: If the slope k of the tangent line at any point P(x₀, y₀) on the graph of y = F(x) (for x in (0, 3]) always satisfies k ≤ 1/2, find the minimum value of the real number a.So, essentially, we need to ensure that F'(x) ≤ 1/2 for all x in (0, 3]. And we need to find the minimal a such that this condition holds. So, the maximum value of F'(x) on (0, 3] must be less than or equal to 1/2, and we need to find the smallest a that enforces this.First, let's recall that F'(x) = (x - a)/x². So, we need (x - a)/x² ≤ 1/2 for all x in (0, 3].Therefore, (x - a)/x² ≤ 1/2.Multiplying both sides by x² (which is positive since x > 0), we get:x - a ≤ (1/2)x².Rearranging terms:- a ≤ (1/2)x² - xMultiply both sides by -1 (remembering to reverse the inequality):a ≥ x - (1/2)x².So, we have a ≥ x - (1/2)x² for all x in (0, 3].Therefore, to satisfy this inequality for all x in (0, 3], a must be greater than or equal to the maximum value of the function h(x) = x - (1/2)x² on the interval (0, 3]. Therefore, the minimal such a is the maximum of h(x) on (0, 3].Hence, our task reduces to finding the maximum of h(x) = x - (1/2)x² on (0, 3]. Then, the minimal a is that maximum.So, let's find the maximum of h(x) = x - (1/2)x² on (0, 3]. To do this, we can take the derivative of h(x):h'(x) = 1 - x.Setting h'(x) = 0 gives 1 - x = 0 => x = 1.So, h(x) has a critical point at x = 1. Since h'(x) is positive when x < 1 (since 1 - x > 0) and negative when x > 1, the function h(x) has a maximum at x = 1.Therefore, the maximum value of h(x) on (0, 3] is h(1) = 1 - (1/2)(1)^2 = 1 - 1/2 = 1/2.But wait, we also need to check the endpoints of the interval. However, the interval is (0, 3], but x approaches 0 from the right and x approaches 3. Let's compute the limits as x approaches 0 and the value at x = 3.As x approaches 0 from the right:lim_{x→0+} h(x) = lim_{x→0+} [x - (1/2)x²] = 0 - 0 = 0.At x = 3:h(3) = 3 - (1/2)(9) = 3 - 4.5 = -1.5.So, the maximum of h(x) on (0, 3] is indeed at x = 1, where h(1) = 1/2. Therefore, the maximum value is 1/2. Hence, the minimal a must be 1/2.But let me verify again. If we set a = 1/2, then for all x in (0, 3], we have:F'(x) = (x - a)/x² = (x - 1/2)/x².We need to confirm that this is ≤ 1/2 for all x in (0, 3].But wait, when a = 1/2, the maximum value of h(x) is 1/2, so a = 1/2 is the minimal a such that a ≥ h(x) for all x in (0, 3]. Therefore, substituting back, the inequality (x - a)/x² ≤ 1/2 becomes equality when x =1, because h(1)=1/2. Therefore, at x =1, (1 -1/2)/1² = 1/2, which is equal to the required slope. For other x, the slope would be less than or equal to 1/2.Wait, but maybe I should check with a=1/2 whether F'(x) ≤ 1/2 for all x in (0,3]. Let's take some test points.Take x approaching 0: F'(x) = (x - 0.5)/x². As x approaches 0 from the right, the numerator approaches -0.5, denominator approaches 0, so the expression approaches -infinity. So, F'(x) approaches negative infinity, which is certainly less than 1/2.At x=1: F'(1) = (1 - 0.5)/1 = 0.5, which is equal to 1/2.At x=3: F'(3) = (3 - 0.5)/9 = 2.5/9 ≈ 0.277..., which is less than 1/2.Now, let's check if there's any other maximum in between. The function h(x) was already maximized at x=1, so with a=1/2, the maximum of F'(x) is 1/2 at x=1, and everywhere else it's less. Therefore, this seems to satisfy the condition.But just to make sure, let's analyze F'(x) when a=1/2. So F'(x) = (x - 1/2)/x².We can also check for other critical points of F'(x). Wait, but we need to ensure that F'(x) ≤ 1/2 for all x in (0,3]. But since we have already found that a=1/2 is the minimal a that satisfies the inequality a ≥ h(x) for all x in (0,3], and h(x) is maximized at x=1 with value 1/2, then a=1/2 is indeed the minimal value. Therefore, the minimal a is 1/2.Wait, but let me make sure there isn't a point where even with a=1/2, F'(x) could exceed 1/2. Let's take the derivative of F'(x) with respect to x to see if there are any local maxima.Wait, F'(x) = (x - a)/x². Let's compute its derivative to check for extrema.F''(x) = derivative of (x - a)/x². Using quotient rule:[(1)(x²) - (x - a)(2x)] / x^4Simplify numerator:x² - 2x(x - a) = x² - 2x² + 2a x = -x² + 2a x.So F''(x) = (-x² + 2a x)/x^4 = (-x + 2a)/x³.Setting F''(x) = 0:(-x + 2a)/x³ = 0 => -x + 2a = 0 => x = 2a.Therefore, F'(x) has a critical point at x = 2a. Since we set a=1/2, then x=2*(1/2)=1. So, F'(x) has a critical point at x=1. Since the second derivative of F'(x) (which is F''(x)) changes sign here, but actually, in this case, since we have F''(x) = (-x + 2a)/x³. At x=1, if a=1/2, then F''(1)= (-1 + 1)/1³=0. Wait, this might be a point of inflection. Wait, perhaps better to check the sign around x=1.Wait, when x approaches 1 from the left (x <1), say x=0.5:F''(0.5)= (-0.5 + 2*(0.5))/ (0.5)^3= (-0.5 +1)/0.125=0.5/0.125=4>0. So concave up.When x approaches 1 from the right (x>1), say x=1.5:F''(1.5)= (-1.5 +1)/ (1.5)^3= (-0.5)/3.375≈-0.148<0. So concave down.Therefore, x=1 is a point of inflection for F'(x). So, the critical point at x=1 for F'(x) is actually a maximum because F''(x) changes from positive to negative, meaning F'(x) changes from concave up to concave down, so x=1 is a local maximum for F'(x).Therefore, when a=1/2, the maximum of F'(x) on (0,3] occurs at x=1, and it's equal to 1/2. Thus, the slope of the tangent line is indeed always less than or equal to 1/2. Therefore, the minimal a is 1/2.But let me test with a slightly larger a. Suppose a=1. Then F'(x)=(x -1)/x². Then at x=1, F'(1)=0, and at x=2, F'(2)=1/4, which is still less than 1/2. But if a is less than 1/2, say a=0.4. Then F'(x)=(x -0.4)/x². At x=1, that would be (1 -0.4)/1=0.6, which is greater than 0.5, violating the condition. Therefore, a must be at least 1/2 to ensure that at x=1, the slope is exactly 1/2, and for all other x, the slope is less. So, yes, the minimal a is 1/2.Thus, after all these steps, the answers should be:(1) F(x) is decreasing on (0, a) and increasing on (a, ∞).(2) The minimum value of a is 1/2.Final Answer(1) The function ( F(x) ) is decreasing on ( (0, a) ) and increasing on ( (a, infty) ). (2) The minimum value of ( a ) is (boxed{dfrac{1}{2}}).</think>Given the functions ( f(x) = ln x ), ( g(x) = frac{a}{x} ) (with ( a > 0 )), and ( F(x) = f(x) + g(x) ):1. To find the intervals of monotonicity for ( F(x) ), we first compute the derivative: [ F'(x) = frac{1}{x} - frac{a}{x^2} = frac{x - a}{x^2} ] The sign of ( F'(x) ) is determined by the numerator ( x - a ). Since ( x^2 > 0 ) for all ( x > 0 ): - When ( x < a ), ( F'(x) < 0 ), so ( F(x) ) is decreasing. - When ( x > a ), ( F'(x) > 0 ), so ( F(x) ) is increasing. Therefore, ( F(x) ) is decreasing on ( (0, a) ) and increasing on ( (a, infty) ).2. To ensure the slope ( k ) of the tangent line at any point on ( y = F(x) ) (for ( x in (0, 3] )) satisfies ( k leq frac{1}{2} ), we need: [ frac{x - a}{x^2} leq frac{1}{2} quad text{for all} quad x in (0, 3] ] Rearranging this inequality: [ x - a leq frac{1}{2}x^2 implies a geq x - frac{1}{2}x^2 ] We need ( a ) to be at least the maximum value of ( h(x) = x - frac{1}{2}x^2 ) on ( (0, 3] ). Finding the critical points of ( h(x) ): [ h'(x) = 1 - x = 0 implies x = 1 ] Evaluating ( h(x) ) at the critical point and endpoints: - ( h(1) = 1 - frac{1}{2}(1)^2 = frac{1}{2} ) - ( lim_{x to 0^+} h(x) = 0 ) - ( h(3) = 3 - frac{1}{2}(9) = -1.5 ) The maximum value of ( h(x) ) on ( (0, 3] ) is ( frac{1}{2} ). Therefore, the minimum value of ( a ) is ( frac{1}{2} ).Final Answers:1. The function ( F(x) ) is decreasing on ( (0, a) ) and increasing on ( (a, infty) ).2. The minimum value of ( a ) is (boxed{dfrac{1}{2}}).]

🔑:Solution: (1) F(x)=ln x+ frac {a}{x}(x > 0), F′(x)= frac {1}{x}- frac {a}{x^{2}}= frac {x-a}{x^{2}}, a > 0, When x > a, F′(x) > 0, f(x) is increasing on (a,+infty), When 0 0(0 < xleqslant 3), k=y′| ;_{ x= x_{0} }= frac {x_{0}-a}{x_{0}^{2}}leqslant frac {1}{2}(0 < x_{0}leqslant 3) always holds Leftrightarrow ageqslant (- frac {1}{2}x_{0}^{2}+x_{0})_{max}, When x_{0}=1, (- frac {1}{2}x_{0}^{2}+x_{0}) reaches its maximum value frac {1}{2}, therefore ageqslant frac {1}{2}, therefore boxed{a_{min}= frac {1}{2}}.

🧠:[<think>Okay, let's see. So the problem is about a diagram where points A, B, and C form a straight angle. Hmm, a straight angle is 180 degrees, so that means points A, B, and C are colinear, right? So like, A---B---C in a straight line. Then there's a triangle ADB that's right-angled at D. So triangle ADB has a right angle at D. Then it says angle DBC is 130 degrees. We need to find the measure of angle DAB.First, let me try to visualize this. A, B, C are in a straight line. So B is between A and C. Then triangle ADB is right-angled at D. So point D must be somewhere off the line AC such that when you connect A to D and D to B, you get a right triangle with the right angle at D. Then angle DBC is 130 degrees. So point D is connected to B and C, making angle at B equal to 130 degrees. Wait, but angle DBC is at point B, between D and C? Wait, DBC would be angle at B, formed by points D, B, and C. So since B is on the line AC, angle DBC is 130 degrees. That might help.Let me sketch a rough diagram mentally. Points A, B, C on a straight line. Point D somewhere above the line, forming triangle ADB right-angled at D. Then connecting D to C, but angle at B between D and C is 130 degrees. Hmm. Wait, angle DBC is 130 degrees. So angle between DB and BC is 130 degrees. But BC is part of the straight line AC, so BC is a straight line. So angle DBC is 130 degrees, which is larger than 90 degrees. So that would mean that point D is on one side of the line BC, making that angle.Since triangle ADB is right-angled at D, angles at D is 90 degrees. So angle ADB is 90 degrees. So in triangle ADB, angle at D is 90, so the sum of angles at A and B in triangle ADB must be 90 degrees. So angle DAB (which is what we need to find) plus angle DBA equals 90 degrees.But how does angle DBC being 130 degrees relate to this? Let's think. Since points A, B, C are colinear, angle ABC is a straight angle, 180 degrees. But angle DBC is 130 degrees. So if we consider point D connected to B, then angle between DB and BC is 130 degrees. Therefore, angle between DB and BA would be the remaining part of the straight angle at B. Since angle ABC is 180 degrees, angle DBC is 130 degrees, so angle ABD would be 180 - 130 = 50 degrees. Wait, angle at B between AB and DB is 50 degrees. So angle ABD is 50 degrees.Wait, angle ABD is part of triangle ADB. In triangle ADB, we have angle at D is 90 degrees, angle at B is 50 degrees, so angle at A (which is angle DAB) would be 180 - 90 - 50 = 40 degrees. So is it 40 degrees?Wait, that seems straightforward. Let me check again. So angle DBC is 130 degrees. Since B is on the line AC, the angles on either side of B must add up to 180 degrees. So angle between DB and BC is 130 degrees, so angle between DB and BA is 180 - 130 = 50 degrees. Therefore, in triangle ADB, angles at A and B are angle DAB and angle DBA, which add up to 90 degrees. Since angle DBA is 50 degrees, angle DAB must be 40 degrees. That seems right.But let me make sure there isn't a different configuration. For example, could point D be on the other side of the line AC? If D were on the other side, then angle DBC might be 130 degrees, but then angle ABD would still need to be calculated. Wait, but the triangle ADB is right-angled at D regardless of which side D is on. So if D is on the other side, angle ABD would still be 180 - 130 = 50 degrees. So same result. So regardless of the position of D, as long as triangle ADB is right-angled at D and angle DBC is 130 degrees, angle DAB would be 40 degrees.But maybe there's a different approach. Let's see. Since angle DBC is 130 degrees, and points B, C are colinear with A, we can consider triangle DBC. Wait, but D is connected to B and C. If angle at B is 130 degrees, and BC is a straight line, then DB is a line from B making 130 degrees with BC. Then, considering triangle ADB is right-angled at D, so AD and DB are perpendicular. Then angle ADB is 90 degrees.Wait, but if we can figure out other angles in triangle ADB. Let's see. Let's denote angle DAB as x. Then angle ABD would be 50 degrees as before. Then in triangle ADB, angles are x (at A), 50 degrees (at B), and 90 degrees (at D). So x + 50 + 90 = 180, so x = 40 degrees. So angle DAB is 40 degrees.Alternatively, maybe using coordinates. Let's assign coordinates to the points to verify. Let’s place point B at the origin (0,0). Since A, B, C are colinear, let's have them on the x-axis. Let’s set point A at (-a, 0), point B at (0,0), and point C at (c,0) for some positive a and c. Then point D is somewhere such that triangle ADB is right-angled at D. Let's denote D as (x, y). Since angle at D is 90 degrees, vectors DA and DB must be perpendicular. Vector DA would be ( -a - x, -y ) and vector DB would be ( -x, -y ). The dot product should be zero:(-a - x)(-x) + (-y)(-y) = 0So (a + x)x + y² = 0But also, angle DBC is 130 degrees. Point D is connected to B(0,0) and C(c,0). So angle at B between D and C is 130 degrees. The angle between vectors BD and BC is 130 degrees. Vector BD is (x, y) and vector BC is (c, 0). The angle between BD and BC is 130 degrees.The formula for the angle θ between two vectors u and v is:cosθ = (u · v) / (|u||v|)So cos(130°) = (x*c + y*0) / (sqrt(x² + y²)*sqrt(c² + 0²))Which simplifies to:cos(130°) = (xc) / (sqrt(x² + y²)*c) = x / sqrt(x² + y²)So cos(130°) = x / sqrt(x² + y²)But from earlier, we have (a + x)x + y² = 0. Let's see:(a + x)x + y² = 0=> ax + x² + y² = 0But x² + y² is the square of the distance from D to B, which is |BD|². So:ax + |BD|² = 0But I'm not sure if this helps. Let's see. Alternatively, from the first equation, we have x / sqrt(x² + y²) = cos(130°). Let's denote sqrt(x² + y²) as |BD|. So x = |BD| cos(130°). Then, substituting into the other equation:a*( |BD| cos(130° ) ) + ( |BD| )² = 0Divide both sides by |BD| (assuming |BD| ≠ 0, which it can't be since D is not at B):a cos(130°) + |BD| = 0So |BD| = -a cos(130° )But cos(130°) is negative because 130° is in the second quadrant. So cos(130°) ≈ -0.6428. Therefore, |BD| = -a*(-0.6428) = a*0.6428. So |BD| is positive, which makes sense.But I'm not sure how this helps us find angle DAB. Maybe this coordinate approach is complicating things. Let's stick to the geometric approach.In triangle ADB, we have angle at D is 90°, angle at B is 50°, so angle at A is 40°. Therefore, angle DAB is 40 degrees.Wait, but let me confirm once more. Since angle DBC is 130°, which is the angle between DB and BC. Since BC is a straight line from B to C, and the angle between DB and BC is 130°, then the angle between DB and BA (which is the other direction from B) is 180° - 130° = 50°. Therefore, in triangle ADB, angle at B is 50°, angle at D is 90°, so angle at A is 40°. Therefore, angle DAB is 40°.Yes, that seems correct. I think the answer is 40 degrees.Final AnswerThe measure of angle D A B is boxed{40} degrees.</think>Given that points (A), (B), and (C) form a straight angle, they are collinear. (triangle ADB) is right-angled at (D), and (angle DBC = 130^circ). We need to find the measure of (angle DAB).1. Since (A), (B), and (C) are collinear, (angle ABC = 180^circ).2. (angle DBC = 130^circ) is given. The angle between (DB) and (BC) is (130^circ).3. The remaining angle at (B) between (DB) and (BA) is (180^circ - 130^circ = 50^circ). Thus, (angle ABD = 50^circ).4. In (triangle ADB), which is right-angled at (D), the sum of the angles at (A) and (B) is (90^circ).5. Therefore, (angle DAB = 90^circ - angle ABD = 90^circ - 50^circ = 40^circ).The measure of (angle DAB) is (boxed{40}) degrees.,<think>Okay, let me try to solve this geometry problem. So, we have points A, B, and C forming a straight angle, which means they are colinear, right? So, points A, B, and C lie on a straight line, making angle ABC a straight angle of 180 degrees. Then, there's triangle ADB, which is right-angled at D. That means angle at D, which is angle ADB, is 90 degrees. Also, we are given that angle DBC is 130 degrees. The question is asking for the measure of angle DAB. Let me try to visualize this. Since A, B, and C are colinear, with B in between A and C. Then, point D is somewhere such that triangle ADB is right-angled at D. Also, angle DBC is 130 degrees. Hmm, so angle DBC is the angle at point B between points D and C. Since angle DBC is 130 degrees, and points A, B, C are colinear, then angle DBC is part of the straight line at B. Wait, angle DBC is 130 degrees, but since ABC is a straight line, the sum of angles around point B on one side should be 180 degrees. Wait, but angle DBC is 130 degrees, so maybe angle ABD is 50 degrees? Because 180 - 130 = 50. Is that right? Let me check. If points A, B, C are colinear, then the straight line AC has point B in between. So, the angle between BD and BC is 130 degrees, so the angle between BD and BA would be 180 - 130 = 50 degrees. So, angle ABD is 50 degrees. But wait, angle ABD is part of triangle ADB. Triangle ADB is right-angled at D, so angle ADB is 90 degrees. So, in triangle ADB, we have angle at D is 90 degrees, angle at B is angle ABD, which we just thought is 50 degrees, and angle at A is angle DAB, which is what we need to find. Since the sum of angles in a triangle is 180 degrees, then angle at A (DAB) + angle at B (ABD) + angle at D (ADB) = 180. So, angle DAB + 50 + 90 = 180. Therefore, angle DAB = 180 - 50 - 90 = 40 degrees. So, is the answer 40 degrees? Wait, but let me make sure I didn't skip any steps or make wrong assumptions.Wait, first, how do we know that angle ABD is 50 degrees? Because angle DBC is 130 degrees. Since points A, B, C are colinear, the angles around point B on the straight line AC should add up to 180 degrees. So, angle between BD and BC is 130 degrees, so the angle between BD and BA must be 180 - 130 = 50 degrees. Therefore, angle ABD is 50 degrees. But in triangle ADB, right-angled at D, so angles at A and B should add up to 90 degrees, right? Because in a right-angled triangle, the two non-right angles are complementary. So, angle at A (DAB) + angle at B (ABD) = 90 degrees. Wait, but earlier I calculated angle DAB as 40 degrees, and angle ABD as 50 degrees, which add up to 90. That seems correct. So, then angle DAB is 40 degrees. But let me check if there's another way to approach this. Maybe using exterior angles or something else. Since angle DBC is 130 degrees, and BD is a line from B to D. If we can find other angles in triangle DBC, maybe. But wait, do we have any information about triangle DBC? Not that I know. We know angle DBC is 130 degrees, but we don't know any sides or other angles. However, since A, B, C are colinear, and D is connected to A and B. Maybe using the straight line and considering supplementary angles. Alternatively, maybe drawing a diagram would help. Let me sketch mentally: points A --- B --- C on a straight line. Point D somewhere such that triangle ADB is right-angled at D. Also, angle DBC is 130 degrees. So, from point B, there's a line BD making 130 degrees with BC. Since BC is going to the right (assuming A is left of B and C is right of B), then BD would be going upwards and to the left, making a 130-degree angle with BC. Wait, angle DBC is 130 degrees, so between BD and BC is 130 degrees. Then, BD is going upwards and to the left from B, creating angle DBC of 130. Then, triangle ADB is right-angled at D, so AD and DB form a right angle at D. Therefore, point D is somewhere above the line AC, connected to A and B, forming a right angle at D. So, in triangle ADB, right-angled at D, with angle at B being 50 degrees (since angle ABD is 50 degrees, as established earlier). Therefore, angle at A is 40 degrees. Wait, but maybe there's a different configuration. What if point D is on the other side of the line AC? But if triangle ADB is right-angled at D, then D has to be positioned such that AD and BD are the legs of the right angle. If angle DBC is 130 degrees, then BD is forming a 130-degree angle with BC, which is part of the straight line. So, BD is going to the left and upwards from B, making 130 degrees with BC. So, in that case, point D is above the line AC, forming triangle ADB with a right angle at D. Alternatively, if D were below the line AC, then angle DBC would be measured differently. But since angle DBC is 130 degrees, which is obtuse, BD has to be on the side where the angle is larger. If D were below the line AC, then angle DBC might be acute, but since it's 130 degrees, which is obtuse, BD is above the line AC. Therefore, with D above AC, forming triangle ADB right-angled at D. Then, angle ABD is 50 degrees, angle ADB is 90 degrees, so angle DAB is 40 degrees. Is there a possibility that angle ABD is not 50 degrees? Let me double-check. Since points A, B, C are colinear, the angle around point B is 180 degrees. The angle between BD and BC is 130 degrees, so the remaining angle between BD and BA is 180 - 130 = 50 degrees. Therefore, angle ABD is indeed 50 degrees. Therefore, in triangle ADB, angles are 90 degrees at D, 50 degrees at B, so angle at A is 40 degrees. Therefore, angle DAB is 40 degrees. Wait, but let me think again. If triangle ADB is right-angled at D, then sides AD and BD are the legs, and AB is the hypotenuse. So, angle at A is opposite to BD, and angle at B is opposite to AD. But maybe using trigonometry here could help. If angle ABD is 50 degrees, then in triangle ABD, we can apply trigonometric ratios. Let me denote angle DAB as x. Then, angle at D is 90, angle at B is 50, angle at A is x. Then, as before, x = 180 - 90 - 50 = 40 degrees. So, angle DAB is 40 degrees. Alternatively, using the sine formula: in triangle ABD, sin(x)/BD = sin(50)/AD. But without knowing the lengths, it's not helpful. However, since it's a right-angled triangle, we know that tan(x) = opposite/adjacent. In triangle ABD, angle at A is x, so the opposite side is BD, and adjacent is AD. So, tan(x) = BD/AD. But we don't have the lengths. But since we know angle at B is 50 degrees, and angle at D is 90, the remaining angle at A must be 40 degrees. Therefore, the answer is 40 degrees. Wait, is there a possibility that the triangle ADB is not on the same side as point C? For example, if D is on the opposite side of AB, then angle DBC could still be 130 degrees. But since A, B, C are colinear with B between A and C, if D is on the opposite side, then angle DBC would be measured from BD to BC, which might be different. Hmm. Wait, if D is on the other side of AB, then the angle DBC would be measured in the other direction. Let's say A is on the left, B in the middle, C on the right. If D is above the line AC, then angle DBC is formed by BD going up and to the left, making 130 degrees with BC (which is towards the right). But if D is below the line AC, then angle DBC would be formed by BD going down and to the left, making 130 degrees with BC. But in either case, the angle between BD and BC is 130 degrees, so the angle between BD and BA would still be 50 degrees, because they are supplementary. Wait, no. If D is below the line AC, then the angle between BD and BC would be 130 degrees, but the angle between BD and BA would be measured on the other side. Let me think. If points A, B, C are on a straight line from left to right, and D is below the line, then angle DBC is the internal angle between BD and BC. If BD is going down and to the left from B, then the angle between BD and BC (which is towards the right) would be 130 degrees. Then, the angle between BD and BA (which is towards the left) would be 180 - 130 = 50 degrees, same as before. So, regardless of whether D is above or below the line AC, the angle between BD and BA would still be 50 degrees. Therefore, angle ABD is 50 degrees. Therefore, in triangle ADB, right-angled at D, angle at B is 50 degrees, angle at D is 90 degrees, so angle at A is 40 degrees. Therefore, angle DAB is 40 degrees. But wait, hold on. If D is below the line AC, then triangle ADB being right-angled at D would mean that AD and BD meet at D below the line. But does that affect the angles? No, because angles are measured the same way regardless of the position. So, the triangle's angles still add up to 180, so angle at A is still 40 degrees. Therefore, regardless of where D is located (above or below the line AC), as long as triangle ADB is right-angled at D and angle DBC is 130 degrees, angle DAB should be 40 degrees. Is there another way this could be approached? Maybe using coordinates. Let's assign coordinates to the points to calculate. Let me place point B at the origin (0,0). Since A, B, C are colinear, let's set them along the x-axis. Let’s say point A is at (-a, 0), point B is at (0,0), and point C is at (c, 0), where a and c are positive numbers. Point D is somewhere such that triangle ADB is right-angled at D. Let's denote the coordinates of D as (x, y). Since triangle ADB is right-angled at D, the vectors DA and DB should be perpendicular. Vector DA is from D to A: (-a - x, -y)Vector DB is from D to B: (-x, -y)Their dot product should be zero because they are perpendicular:(-a - x)(-x) + (-y)(-y) = 0(a + x)x + y² = 0ax + x² + y² = 0Additionally, angle DBC is 130 degrees. Let's figure out the coordinates part. Angle DBC is the angle at point B between points D and C. Since point B is at (0,0), point C is at (c, 0), and point D is at (x, y). The angle between vectors BD and BC is 130 degrees. Vector BD is D - B = (x, y)Vector BC is C - B = (c, 0)The angle between vectors BD and BC is 130 degrees. The formula for the angle between two vectors is:cosθ = (BD · BC) / (|BD| |BC|)So,cos(130°) = (x * c + y * 0) / (sqrt(x² + y²) * sqrt(c² + 0²))cos(130°) = (c x) / (sqrt(x² + y²) * c)cos(130°) = x / sqrt(x² + y²)Therefore,x / sqrt(x² + y²) = cos(130°)Let me compute cos(130°). 130° is in the second quadrant, so cos(130°) = -cos(50°) ≈ -0.6428Therefore,x / sqrt(x² + y²) ≈ -0.6428So, x is negative because the right side is negative. Therefore, point D has a negative x-coordinate, which makes sense because angle DBC is 130°, so BD is going towards the left from B. So, from point B at (0,0), BD is going to the left (negative x-direction) and some y-direction (up or down). But also, we have the equation from the right angle at D:ax + x² + y² = 0Hmm. Let's see if we can solve these equations. From the angle condition:x = -0.6428 sqrt(x² + y²)Square both sides:x² = 0.4132 (x² + y²)x² = 0.4132 x² + 0.4132 y²x² - 0.4132 x² = 0.4132 y²0.5868 x² = 0.4132 y²Divide both sides by 0.4132:(0.5868 / 0.4132) x² = y²Approximately:1.419 x² = y²So, y = ±√1.419 x ≈ ±1.191 xNow, plug this into the other equation:ax + x² + y² = 0Substitute y² ≈ 1.419 x²:ax + x² + 1.419 x² = 0ax + (1 + 1.419) x² = 0ax + 2.419 x² = 0Factor out x:x(a + 2.419 x) = 0So, either x = 0 or a + 2.419 x = 0x = 0 would mean point D is at (0, y), but then triangle ADB would be right-angled at D, which is at (0,y). Then, points A(-a,0), D(0,y), B(0,0). Then, vectors DA = (-a, -y) and DB = (0, -y). Their dot product is (-a)(0) + (-y)(-y) = y². For them to be perpendicular, the dot product should be zero, so y² = 0, which implies y=0. But that would make D coincide with B, which is impossible. Therefore, x=0 is not a valid solution. Therefore, we must have:a + 2.419 x = 0Therefore,x = -a / 2.419 ≈ -0.4132 aSo, x ≈ -0.4132 aThen, y ≈ ±1.191 x ≈ ±1.191 (-0.4132 a) ≈ ∓0.492 aSo, point D is at approximately (-0.4132 a, ±0.492 a). Since the problem doesn't specify the position of D above or below, but angle DBC is 130°, which is obtuse, so if D is below the line AC, the angle would still be 130°, so both positions are possible. Now, we need to find angle DAB. Let's compute this angle using coordinates. Point A is at (-a, 0)Point D is at (-0.4132 a, ±0.492 a)Point B is at (0,0)We need angle at A, which is angle DAB. To find angle DAB, we can compute the angle between vectors AD and AB. Vector AD is from A to D: (-0.4132 a - (-a), ±0.492 a - 0) = (0.5868 a, ±0.492 a)Vector AB is from A to B: (0 - (-a), 0 - 0) = (a, 0)The angle between vectors AD and AB can be found using the dot product formula:cosθ = (AD · AB) / (|AD| |AB|)Compute AD · AB:(0.5868 a)(a) + (±0.492 a)(0) = 0.5868 a²|AD| = sqrt((0.5868 a)^2 + (±0.492 a)^2) = sqrt(0.3443 a² + 0.2421 a²) = sqrt(0.5864 a²) ≈ 0.7658 a|AB| = sqrt(a² + 0) = aTherefore,cosθ = 0.5868 a² / (0.7658 a * a) = 0.5868 / 0.7658 ≈ 0.766Therefore, θ ≈ arccos(0.766) ≈ 40 degreesSo, angle DAB is approximately 40 degrees. Therefore, the answer is 40 degrees. This corroborates our earlier reasoning. Therefore, the measure of angle DAB is 40 degrees.Final AnswerThe measure of angle D A B is boxed{40} degrees.</think>Given that points (A), (B), and (C) form a straight angle, they are collinear. Triangle (ADB) is right-angled at (D), and (angle DBC = 130^circ). We need to find the measure of (angle DAB).1. Identify Supplementary Angles: - Since (A), (B), and (C) are collinear, (angle ABC = 180^circ). - (angle DBC = 130^circ) is given. The supplementary angle to (angle DBC) at point (B) is (angle ABD). - Therefore, (angle ABD = 180^circ - 130^circ = 50^circ).2. Sum of Angles in Triangle (ADB): - Triangle (ADB) is right-angled at (D), so (angle ADB = 90^circ). - The sum of the angles in triangle (ADB) is (180^circ). - Let (angle DAB = x). Then, (x + 50^circ + 90^circ = 180^circ). - Solving for (x), we get (x = 180^circ - 50^circ - 90^circ = 40^circ).3. Verification with Coordinates: - Assign coordinates to points (A), (B), and (C) on the x-axis. - Use vector analysis and dot product to confirm the right angle at (D). - Calculate the angle (angle DAB) using vector dot product, confirming it is (40^circ).Thus, the measure of (angle DAB) is (boxed{40}) degrees.]

🔑:Since A B C is a straight angle, then angle D B A=180^{circ}-angle D B C=180^{circ}-130^{circ}=50^{circ}.Since the angles in a triangle add to 180^{circ}, thenangle D A B=180^{circ}-angle A D B-angle D B A=180^{circ}-90^{circ}-50^{circ}=40^{circ}(Alternatively, we could note that angle D B C is an exterior angle for the triangle.This means that angle D B C=angle D A B+angle A D B, which gives angle D A B=angle D B C-angle A D B= 130^{circ}-90^{circ}=40^{circ}.ANSWER: 40^{circ}