🧠:[<think>Okay, so I need to find the value of 'a' such that the line y = 5x + a passes through the point (a, a²). Hmm, let's start by understanding what this means. The line equation is given as y = 5x + a. That's a linear equation in slope-intercept form, where the slope is 5 and the y-intercept is 'a'. The problem states that this line passes through the specific point (a, a²). Since the point (a, a²) lies on the line, its coordinates should satisfy the equation of the line. So, if I substitute x = a and y = a² into the equation y = 5x + a, the equation should hold true. Let me write that out:a² = 5(a) + aSimplifying the right side:a² = 5a + aa² = 6aNow, I need to solve for 'a'. Let's bring all terms to one side to form a quadratic equation:a² - 6a = 0Factor out an 'a':a(a - 6) = 0This gives two solutions: a = 0 or a = 6. But the problem specifies that a ≠ 0, so the only valid solution is a = 6. Wait, let me check that again to make sure I didn't make a mistake. If a = 0, then the original line equation becomes y = 5x, and the point would be (0, 0), which does lie on that line. But the problem says a ≠ 0, so we have to exclude that solution. Therefore, a must be 6. Let me verify by plugging a = 6 back into the equation. The line becomes y = 5x + 6. The point would be (6, 6²) which is (6, 36). Plugging x = 6 into the line equation: y = 5*6 + 6 = 30 + 6 = 36. Yep, that matches the y-coordinate of the point. So that works. Hmm, seems straightforward. Is there any other possible solution? Well, the quadratic equation only factors into a(a - 6) = 0, so unless there's a mistake in the algebra, those are the only solutions. Since we're told a ≠ 0, 6 is the answer. I think that's it. Maybe the problem is testing the understanding of substituting points into equations and solving quadratics. Alright, confident enough that a = 6 is the right answer.Final AnswerThe value of ( a ) is boxed{6}.</think>To find the value of ( a ) such that the line ( y = 5x + a ) passes through the point ( (a, a^2) ), we substitute ( x = a ) and ( y = a^2 ) into the equation of the line:[a^2 = 5a + a]Simplifying the right side, we get:[a^2 = 6a]Rearranging into a quadratic equation:[a^2 - 6a = 0]Factoring out ( a ):[a(a - 6) = 0]This gives two solutions: ( a = 0 ) or ( a = 6 ). Since ( a neq 0 ), the valid solution is:[a = 6]Verification: Substituting ( a = 6 ) into the line equation ( y = 5x + 6 ) and the point ( (6, 36) ) shows that the point lies on the line, confirming the solution is correct.The value of ( a ) is (boxed{6}).,<think>Okay, let's see. I need to find the value of 'a' such that the line with equation y = 5x + a passes through the point (a, a²). Hmm, alright. Let me try to break this down step by step.First, I know that if a point (x, y) lies on a line, then plugging the x and y values into the equation of the line should satisfy it. That means if the point (a, a²) is on the line y = 5x + a, then substituting x = a and y = a² into the equation should make both sides equal.So let me do that substitution. The original equation is y = 5x + a. When x is 'a', y should be 5 times 'a' plus 'a', right? But wait, the point is (a, a²), so the y-coordinate is a². Therefore, substituting x = a into the equation gives y = 5a + a. But that y has to equal a². So putting it all together:a² = 5a + aHmm, simplifying the right side: 5a + a is 6a. So now the equation becomes:a² = 6aNow, to solve for 'a', I can subtract 6a from both sides to set the equation to zero:a² - 6a = 0Factoring out an 'a' from the left side:a(a - 6) = 0This gives two solutions: a = 0 or a = 6. But wait, the problem states that a ≠ 0, so the only valid solution is a = 6. Let me check if that makes sense.If a = 6, then the line equation is y = 5x + 6. The point given is (a, a²) which would be (6, 36). Let's plug x = 6 into the line equation: y = 5*6 + 6 = 30 + 6 = 36. That matches the y-coordinate of the point (6, 36). So that works.Just to make sure I didn't make a mistake earlier, let's verify the original equation. When a = 6, substituting into the line equation y = 5x + 6, the point (6, 36) is on the line because 36 = 5*6 + 6 = 30 + 6. Yep, that's correct. And since a ≠ 0 is given, a = 6 is the answer.Wait, but let me double-check my substitution step. The line equation is y = 5x + a. The point is (a, a²). So when x is 'a', y should be 5a + a. Wait, hold on. The equation is y = 5x + a. So substituting x = a, then y = 5*a + a. Wait, is that correct?Wait, no. Wait, the equation is y = 5x + a, so if x is a, then y is 5a + a. But a is also the parameter in the line equation. So here, both the line's y-intercept and the x-coordinate of the point are 'a'. So in this substitution, you substitute x with 'a', then compute y as 5 times 'a' plus 'a', which is 6a, and that must equal the y-coordinate of the point, which is a². So a² = 6a, leading to a² - 6a = 0, so a(a - 6) = 0, so a = 0 or 6. Since a ≠ 0, the answer is 6. That seems right.But just to make sure there's no trick here. The line is y = 5x + a. The line's y-intercept is 'a', and the point we're checking is (a, a²). So substituting x = a gives y = 5a + a. So y = 6a. Since the point is (a, a²), that y must equal a². Therefore, 6a = a². Subtract 6a: a² - 6a = 0. Factor: a(a - 6) = 0. Solutions a = 0 or 6. Exclude 0, so a = 6. Yep, that seems solid. I think that's correct. So the answer is 6.Final AnswerThe value of ( a ) is boxed{6}.</think>To find the value of ( a ) such that the line ( y = 5x + a ) passes through the point ( (a, a^2) ), we substitute ( x = a ) and ( y = a^2 ) into the line equation:[a^2 = 5a + a]Simplifying the right side, we get:[a^2 = 6a]Rearranging the equation to set it to zero:[a^2 - 6a = 0]Factoring out ( a ):[a(a - 6) = 0]This gives the solutions ( a = 0 ) or ( a = 6 ). Since ( a neq 0 ), the valid solution is:[a = 6]Verifying, if ( a = 6 ), the line equation is ( y = 5x + 6 ). The point ( (6, 36) ) satisfies this equation because ( 36 = 5 cdot 6 + 6 = 30 + 6 ).Thus, the value of ( a ) is (boxed{6}).]

🔑:Since left(a, a^{2}right) lies on the line with equation y=5 x+a, then a^{2}=5 a+a or a^{2}=6 a.Since a neq 0, then a=6.

🧠:[<think>Okay, so I need to find the angle between edges AD and BC in a tetrahedron ABCD. The given ratios are BD:CD:AB:AC:AD:BC = √3 : √2 : 1 : 1 : 1 : 1. Hmm, that's a bit confusing. Let me try to parse this.First, the problem states the ratios between the lengths of these edges: BD, CD, AB, AC, AD, and BC are √3, √2, 1, 1, 1, 1 respectively. Wait, actually, the colon notation usually means the ratio. So BD : CD : AB : AC : AD : BC = √3 : √2 : 1 : 1 : 1 : 1. So each of these edges has a length proportional to those values. So BD is √3 times some unit, CD is √2 times that same unit, and AB, AC, AD, BC are all 1 times that unit. So if we let the unit be 'k', then BD = √3 k, CD = √2 k, AB = k, AC = k, AD = k, BC = k.But maybe we can just assume the unit is 1 for simplicity? Let me check. If we set k = 1, then BD = √3, CD = √2, AB = 1, AC = 1, AD = 1, BC = 1. So all those edges have those lengths. So the tetrahedron has edges AB, AC, AD, BC all of length 1, BD is √3, CD is √2. Let me visualize this.A tetrahedron has four triangular faces. So vertices A, B, C, D. Let's list the edges:AB = 1, AC = 1, AD = 1, BC = 1, BD = √3, CD = √2. So the edges connected to A are AB, AC, AD, all length 1. Then edges BC is 1, BD is √3, CD is √2. So maybe vertex A is connected to B, C, D each with length 1. Then between B and C is length 1, B and D is √3, C and D is √2. So the face BCD has sides BC = 1, BD = √3, CD = √2.Wait, but in a tetrahedron, each pair of vertices is connected by an edge, so all six edges are present. So in this case, the edges AB, AC, AD, BC, BD, CD are all present with the given lengths. So perhaps we can assign coordinates to the points and then compute the angle between AD and BC.The angle between two edges in space can be found using the dot product formula. The angle θ between vectors u and v is given by cosθ = (u · v)/(|u| |v|). But here, the edges AD and BC are skew lines, right? Because in a tetrahedron, edges that don't share a common vertex are skew. So AD and BC don't intersect and aren't parallel. Wait, but how do we find the angle between two skew lines?I remember that the angle between two skew lines is defined as the angle between their direction vectors when projected onto a plane perpendicular to both. Alternatively, you can translate one of the vectors so that they intersect, and then find the angle between them. But since they are skew, they don't intersect, so the angle is determined by the direction vectors.Wait, perhaps the angle between two skew lines is the angle between their direction vectors. So maybe even though they are skew, the angle between them is just the angle between the vectors AD and BC. So if I can find vectors along AD and BC, compute their dot product, and then divide by the product of their magnitudes, that should give the cosine of the angle between them.But let me confirm this. If two lines are skew, the angle between them is still defined as the angle between their direction vectors. Yes, according to my notes, the angle between two skew lines is the angle between their direction vectors, which can be found using the dot product. So even if the lines are skew, as long as we have their direction vectors, the angle is computed the same way as for intersecting lines.Therefore, if I can assign coordinates to the tetrahedron such that I can determine the vectors AD and BC, then compute the angle between them.So the plan is:1. Assign coordinates to points A, B, C, D such that the given edge lengths are satisfied.2. Find the coordinates of these points.3. Determine the vectors AD and BC.4. Compute the dot product of these vectors.5. Divide by the product of their magnitudes to get cosine of the angle.6. Take arccos to find the angle.Let me proceed step by step.First, assign coordinates. Let me place point A at the origin to simplify calculations. So A = (0, 0, 0). Then, since edges AB, AC, AD are all length 1, I can assign points B, C, D somewhere on the unit sphere around A. However, the positions of B, C, D must also satisfy the other edge lengths: BC = 1, BD = √3, CD = √2.Let me try to set coordinates for B, C, D such that:- AB = 1: Let’s place B along the x-axis. So B = (1, 0, 0).- AC = 1: Let’s place C in the xy-plane. Let’s say C = (x, y, 0). Since AC = 1, then x² + y² = 1.- BC = 1: The distance between B (1,0,0) and C (x,y,0) is 1. So sqrt[(x - 1)^2 + (y - 0)^2] = 1. Therefore, (x - 1)^2 + y^2 = 1.Since x² + y² = 1 (from AC = 1), substituting into the BC equation:(x - 1)^2 + (1 - x²) = 1Expanding (x - 1)^2: x² - 2x + 1 + 1 - x² = 1Wait, wait. Let's substitute y² = 1 - x² into (x - 1)^2 + y² = 1:(x - 1)^2 + (1 - x²) = 1Expand (x² - 2x + 1) + 1 - x² = 1Simplify: x² - 2x + 1 + 1 - x² = (-2x + 2) = 1So -2x + 2 = 1 → -2x = -1 → x = 1/2Then y² = 1 - x² = 1 - 1/4 = 3/4 → y = ±√3/2. Let's pick y = √3/2 to place C in the upper half-plane. So C = (1/2, √3/2, 0).So now, points A = (0,0,0), B = (1,0,0), C = (1/2, √3/2, 0). Now we need to find coordinates for D such that AD = 1, BD = √3, CD = √2.Since AD = 1, point D is somewhere on the unit sphere around A, so D = (d_x, d_y, d_z) with d_x² + d_y² + d_z² = 1.Then BD = √3: The distance between B (1,0,0) and D (d_x, d_y, d_z) is √3. Therefore,(d_x - 1)^2 + (d_y - 0)^2 + (d_z - 0)^2 = 3.Similarly, CD = √2: The distance between C (1/2, √3/2, 0) and D is √2:(d_x - 1/2)^2 + (d_y - √3/2)^2 + (d_z - 0)^2 = 2.So we have three equations:1. d_x² + d_y² + d_z² = 1 (AD = 1)2. (d_x - 1)^2 + d_y² + d_z² = 3 (BD = √3)3. (d_x - 1/2)^2 + (d_y - √3/2)^2 + d_z² = 2 (CD = √2)Let me expand equations 2 and 3.Equation 2:(d_x - 1)^2 + d_y² + d_z² = 3Expand (d_x² - 2d_x + 1) + d_y² + d_z² = 3But from equation 1, d_x² + d_y² + d_z² = 1, so substitute:1 - 2d_x + 1 = 3 → -2d_x + 2 = 3 → -2d_x = 1 → d_x = -1/2So d_x is -1/2.Then, equation 3:(d_x - 1/2)^2 + (d_y - √3/2)^2 + d_z² = 2We already know d_x = -1/2, so substitute:(-1/2 - 1/2)^2 + (d_y - √3/2)^2 + d_z² = 2Simplify:(-1)^2 + (d_y - √3/2)^2 + d_z² = 21 + (d_y - √3/2)^2 + d_z² = 2 → (d_y - √3/2)^2 + d_z² = 1But from equation 1, with d_x = -1/2:(-1/2)^2 + d_y² + d_z² = 1 → 1/4 + d_y² + d_z² = 1 → d_y² + d_z² = 3/4So we have:From equation 3: (d_y - √3/2)^2 + d_z² = 1Expand (d_y - √3/2)^2: d_y² - √3 d_y + 3/4So substituting into equation 3:d_y² - √3 d_y + 3/4 + d_z² = 1But from d_y² + d_z² = 3/4, replace d_y² + d_z² with 3/4:3/4 - √3 d_y + 3/4 = 1Combine terms:6/4 - √3 d_y = 1 → 3/2 - √3 d_y = 1 → -√3 d_y = -1/2 → d_y = (1/2)/√3 = 1/(2√3) = √3/6Then, from d_y² + d_z² = 3/4:( (√3/6)^2 ) + d_z² = 3/4 → (3/36) + d_z² = 3/4 → 1/12 + d_z² = 3/4 → d_z² = 3/4 - 1/12 = 9/12 - 1/12 = 8/12 = 2/3 → d_z = ±√(2/3) = ±(√6)/3So point D has coordinates (-1/2, √3/6, ±√6/3). Since the tetrahedron is in 3D space, there are two possible positions for D, mirrored over the xy-plane. However, the angle between AD and BC should be the same regardless of the choice of sign for d_z. Let's pick the positive z-coordinate for simplicity, so D = (-1/2, √3/6, √6/3).Now, we have all coordinates:A = (0,0,0)B = (1,0,0)C = (1/2, √3/2, 0)D = (-1/2, √3/6, √6/3)Next, we need to find vectors AD and BC to compute the angle between them.First, vector AD is from A to D: D - A = (-1/2, √3/6, √6/3)Vector BC is from B to C: C - B = (1/2 - 1, √3/2 - 0, 0 - 0) = (-1/2, √3/2, 0)Now, we can compute the dot product of vectors AD and BC.AD · BC = (-1/2)(-1/2) + (√3/6)(√3/2) + (√6/3)(0)Compute each component:First component: (-1/2)(-1/2) = 1/4Second component: (√3/6)(√3/2) = (3/12) = 1/4Third component: (√6/3)(0) = 0Total dot product: 1/4 + 1/4 + 0 = 1/2Now, compute the magnitudes of AD and BC.|AD|: Since AD is length 1, as given. Wait, yes, the problem states AD = 1, so the magnitude is 1.Wait, but let's confirm with the coordinates:|AD| = sqrt[ (-1/2)^2 + (√3/6)^2 + (√6/3)^2 ]Compute each term:(-1/2)^2 = 1/4(√3/6)^2 = (3)/36 = 1/12(√6/3)^2 = 6/9 = 2/3Sum: 1/4 + 1/12 + 2/3 = 3/12 + 1/12 + 8/12 = 12/12 = 1. So yes, |AD| = 1.|BC|: The length BC is given as 1. Let's confirm:Vector BC is (-1/2, √3/2, 0)|BC| = sqrt[ (-1/2)^2 + (√3/2)^2 + 0^2 ] = sqrt[ 1/4 + 3/4 ] = sqrt[1] = 1. Correct.So, the dot product is 1/2, |AD| = 1, |BC| = 1.Thus, cosθ = (1/2)/(1*1) = 1/2.Therefore, the angle θ is arccos(1/2) = 60 degrees, or π/3 radians.Wait, that seems straightforward. But let me check once again to make sure I didn't make any mistakes in calculations.Coordinates:A at (0,0,0)B at (1,0,0)C at (1/2, √3/2, 0)D at (-1/2, √3/6, √6/3)Vectors:AD = D - A = (-1/2, √3/6, √6/3)BC = C - B = (-1/2, √3/2, 0)Dot product:(-1/2)(-1/2) + (√3/6)(√3/2) + (√6/3)(0) = 1/4 + (3/12) + 0 = 1/4 + 1/4 = 1/2Magnitudes:|AD| = 1 (confirmed)|BC| = 1 (confirmed)Thus, cosθ = 1/2 ⇒ θ = 60°.So the angle between AD and BC is 60 degrees, which is π/3 radians.But let me just visualize if this makes sense. In the coordinate system, AD is going from the origin to (-1/2, √3/6, √6/3), which is in the direction of negative x, slightly positive y, and positive z. BC is going from B to C, which is in the direction of negative x, positive y, and no z-component. The angle between them being 60 degrees seems plausible.Alternatively, since both vectors have a magnitude of 1, and their dot product is 1/2, which directly gives the angle as 60 degrees. So I think this is correct.Therefore, the angle between AD and BC is 60 degrees, or π/3 radians.Final AnswerThe angle between ( AD ) and ( BC ) is (boxed{60^circ}).</think>To find the angle between edges (AD) and (BC) in tetrahedron (ABCD) with given edge length ratios (BD: CD: AB: AC: AD: BC = sqrt{3}: sqrt{2}: 1: 1: 1: 1), we can follow these steps:1. Assign Coordinates: - Place (A) at the origin: (A = (0, 0, 0)). - Place (B) along the x-axis: (B = (1, 0, 0)). - Place (C) in the xy-plane: (C = left(frac{1}{2}, frac{sqrt{3}}{2}, 0right)). - Determine coordinates for (D) using the given lengths: - (AD = 1), so (D) lies on the unit sphere. - Solve for (D) using distances (BD = sqrt{3}) and (CD = sqrt{2}): - (D = left(-frac{1}{2}, frac{sqrt{3}}{6}, frac{sqrt{6}}{3}right)).2. Determine Vectors: - Vector (AD) is from (A) to (D): (overrightarrow{AD} = left(-frac{1}{2}, frac{sqrt{3}}{6}, frac{sqrt{6}}{3}right)). - Vector (BC) is from (B) to (C): (overrightarrow{BC} = left(-frac{1}{2}, frac{sqrt{3}}{2}, 0right)).3. Compute Dot Product and Magnitudes: - Dot product: (overrightarrow{AD} cdot overrightarrow{BC} = left(-frac{1}{2}right)left(-frac{1}{2}right) + left(frac{sqrt{3}}{6}right)left(frac{sqrt{3}}{2}right) + left(frac{sqrt{6}}{3}right)(0) = frac{1}{4} + frac{1}{4} + 0 = frac{1}{2}). - Magnitudes: (|overrightarrow{AD}| = 1) and (|overrightarrow{BC}| = 1).4. Calculate the Angle: - Using the dot product formula: (cos theta = frac{overrightarrow{AD} cdot overrightarrow{BC}}{|overrightarrow{AD}| |overrightarrow{BC}|} = frac{1/2}{1 cdot 1} = frac{1}{2}). - Therefore, (theta = arccosleft(frac{1}{2}right) = 60^circ).The angle between (AD) and (BC) is (boxed{60^circ}).,<think>Okay, let's try to tackle this tetrahedron angle problem. So, we have tetrahedron ABCD with some given edge length ratios. The question is asking for the angle between edges AD and BC. Hmm, angles between edges in a tetrahedron... I remember that to find the angle between two lines in 3D space, we can use the dot product formula. The cosine of the angle between two vectors is equal to the dot product of the vectors divided by the product of their magnitudes. But first, we need to figure out the vectors corresponding to AD and BC. Given the ratios of the edges: BD : CD : AB : AC : AD : BC = √3 : √2 : 1 : 1 : 1 : 1. Let me parse that. So BD is √3, CD is √2, AB is 1, AC is 1, AD is 1, and BC is 1. Wait, but ratios are usually given in order. Let me check the original problem again. It says BD:CD:AB:AC:AD:BC = √3:√2:1:1:1:1. So BD is √3, CD is √2, AB is 1, AC is 1, AD is 1, BC is 1. So BD=√3, CD=√2, AB=1, AC=1, AD=1, BC=1. So, the edges BD and CD are longer, while the others are 1. So, we need to find the angle between AD and BC. Since these are edges in a tetrahedron, they might be skew lines, but the angle between skew lines is defined as the angle between their direction vectors. So, even if they don't intersect, we can find the angle by taking the dot product of their direction vectors.But to compute that, we need coordinates for the points. Maybe we can assign coordinates to the vertices and then compute the vectors. Let me try to set up a coordinate system. Let's pick a point to be the origin. Let's choose point A as the origin for simplicity. Then, we can assign coordinates to the other points.Let me denote the coordinates as follows:- Let A be at (0, 0, 0).- Since AB is 1, let's place point B at (1, 0, 0).- AC is 1, so point C can be placed somewhere in the plane. Let me think. If AC is 1, and we need BC to be 1 as well. Since AB is 1 and BC is 1, triangle ABC might be an equilateral triangle? Wait, but AC is 1, AB is 1, BC is 1. So yes, triangle ABC is equilateral. So, point C should be at (0.5, √3/2, 0) to make ABC an equilateral triangle in the xy-plane. Wait, but in 3D space, we can have point C anywhere such that AC=1 and BC=1. But if I place it in the xy-plane, that might simplify things. Let me confirm. If we place A at (0,0,0), B at (1,0,0), then point C in the xy-plane at (0.5, √3/2, 0). Then AB=1, AC=1, BC=1, which fits. Okay, that works.Now, we need to place point D such that AD=1, BD=√3, CD=√2. So point D is somewhere in 3D space. Let's denote D's coordinates as (x, y, z). Then:- AD=1: The distance from A(0,0,0) to D(x,y,z) is √(x² + y² + z²) = 1. So x² + y² + z² = 1.- BD=√3: The distance from B(1,0,0) to D(x,y,z) is √((x−1)² + y² + z²) = √3. Squaring both sides: (x−1)² + y² + z² = 3.- CD=√2: The distance from C(0.5, √3/2, 0) to D(x,y,z) is √((x−0.5)² + (y − √3/2)² + z²) = √2. Squaring: (x−0.5)² + (y − √3/2)² + z² = 2.So we have three equations:1. x² + y² + z² = 12. (x−1)² + y² + z² = 33. (x−0.5)² + (y − √3/2)² + z² = 2Let's expand these equations and solve for x, y, z.Starting with equation 1 and 2:Equation 2: (x−1)² + y² + z² = 3Expand (x−1)²: x² - 2x + 1 + y² + z² = 3But from equation 1, x² + y² + z² = 1. Substitute that in:1 - 2x + 1 = 3So 2 - 2x = 3Therefore, -2x = 1 => x = -0.5So x is found to be -0.5.Now plug x = -0.5 into equation 1 and 3.Equation 1: (-0.5)² + y² + z² = 1 => 0.25 + y² + z² = 1 => y² + z² = 0.75Equation 3: (x−0.5)² + (y − √3/2)² + z² = 2Substituting x = -0.5:(-0.5 - 0.5)² + (y - √3/2)² + z² = 2(-1)² + (y - √3/2)² + z² = 21 + (y - √3/2)² + z² = 2(y - √3/2)² + z² = 1But from equation 1, we have y² + z² = 0.75. Let's write both:1. y² + z² = 0.752. (y - √3/2)² + z² = 1Subtract equation 1 from equation 2:(y - √3/2)² + z² - (y² + z²) = 1 - 0.75Expand (y - √3/2)²: y² - √3 y + (3/4)So, y² - √3 y + 3/4 + z² - y² - z² = 0.25Simplify: -√3 y + 3/4 = 0.25Then: -√3 y = 0.25 - 3/4 = -0.5Multiply both sides by -1: √3 y = 0.5Thus, y = 0.5 / √3 = (√3)/6 ≈ 0.2887So y = √3 / 6.Now, substitute y back into equation 1 to find z.Equation 1: y² + z² = 0.75Compute y²: ( (√3)/6 )² = 3/36 = 1/12 ≈ 0.0833Thus, z² = 0.75 - 1/12 = 9/12 - 1/12 = 8/12 = 2/3Therefore, z = ±√(2/3) = ±(√6)/3 ≈ ±0.8165So, coordinates of D are (-0.5, √3/6, ±√6/3). Hmm, so two possible positions for D, symmetric above and below the xy-plane. Since the problem doesn't specify orientation, but angles are direction-based, perhaps the angle will be the same regardless of the sign of z. Let's pick z = √6/3 for simplicity.Therefore, coordinates of D are (-0.5, √3/6, √6/3).Now, we need to find the angle between AD and BC.First, let's write vectors AD and BC.Vector AD is from A(0,0,0) to D(-0.5, √3/6, √6/3). So the vector is (-0.5, √3/6, √6/3).Vector BC is from B(1,0,0) to C(0.5, √3/2, 0). So the vector is (0.5 - 1, √3/2 - 0, 0 - 0) = (-0.5, √3/2, 0).Alternatively, since the angle between two lines is determined by their direction vectors, regardless of direction. So vectors AD and BC can be taken as above.Now, to find the angle θ between vectors AD and BC, we use the dot product formula:cosθ = (AD · BC) / (|AD| |BC|)First, compute the dot product AD · BC.AD = (-0.5, √3/6, √6/3)BC = (-0.5, √3/2, 0)Dot product = (-0.5)(-0.5) + (√3/6)(√3/2) + (√6/3)(0)Compute each term:First term: (-0.5)(-0.5) = 0.25Second term: (√3/6)(√3/2) = (3)/(12) = 1/4Third term: 0So total dot product = 0.25 + 0.25 + 0 = 0.5Now compute |AD| and |BC|.|AD| is the length of AD, which is given as 1. Since AD is one of the edges with ratio 1, so |AD|=1.Wait, but let's confirm:Compute |AD|: sqrt[ (-0.5)^2 + (√3/6)^2 + (√6/3)^2 ]Calculate each component:(-0.5)^2 = 0.25(√3/6)^2 = (3)/(36) = 1/12 ≈ 0.0833(√6/3)^2 = (6)/(9) = 2/3 ≈ 0.6667Sum: 0.25 + 0.0833 + 0.6667 ≈ 1. So yes, |AD| = 1. Good.|BC|: Length of BC. BC is given as 1. But let's compute it from the vector BC.Vector BC is (-0.5, √3/2, 0). So |BC| = sqrt[ (-0.5)^2 + (√3/2)^2 + 0^2 ] = sqrt[0.25 + 0.75] = sqrt[1] = 1. Perfect.Therefore, cosθ = 0.5 / (1 * 1) = 0.5Therefore, θ = arccos(0.5) = 60 degrees, or π/3 radians.Wait, but let me double-check. The dot product is 0.5, which gives cosθ = 0.5, so θ = 60°. That seems straightforward. But let me check again.AD vector: (-0.5, √3/6, √6/3). BC vector: (-0.5, √3/2, 0). Their dot product is:(-0.5)(-0.5) = 0.25(√3/6)(√3/2) = (3)/(12) = 0.25(√6/3)(0) = 0Total 0.25 + 0.25 = 0.5. Correct.Magnitudes are both 1, so yes, cosθ = 0.5, θ=60°.So, the angle between AD and BC is 60 degrees, which is π/3 radians.But let me make sure that there isn't a mistake in the coordinate setup. For instance, did I assign the coordinates correctly?Points A(0,0,0), B(1,0,0), C(0.5, √3/2, 0), which makes ABC an equilateral triangle with sides of length 1. Then point D is at (-0.5, √3/6, √6/3). Let me confirm the distances:AD: sqrt[ (-0.5)^2 + (√3/6)^2 + (√6/3)^2 ] = sqrt[0.25 + 1/12 + 2/3] = sqrt[0.25 + 0.0833 + 0.6667] = sqrt[1] = 1. Correct.BD: Distance from B(1,0,0) to D(-0.5, √3/6, √6/3):sqrt[ (-0.5 -1)^2 + (√3/6 -0)^2 + (√6/3 -0)^2 ] = sqrt[ (-1.5)^2 + (√3/6)^2 + (√6/3)^2 ]Compute each term:(-1.5)^2 = 2.25(√3/6)^2 = 3/36 = 0.0833(√6/3)^2 = 6/9 = 0.6667Sum: 2.25 + 0.0833 + 0.6667 ≈ 3. So sqrt(3) ≈ 1.732, which matches BD=√3. Correct.CD: Distance from C(0.5, √3/2, 0) to D(-0.5, √3/6, √6/3):sqrt[ (-0.5 -0.5)^2 + (√3/6 - √3/2)^2 + (√6/3 -0)^2 ]Compute each component:x: -1, so squared: 1y: √3/6 - √3/2 = √3/6 - 3√3/6 = (-2√3)/6 = -√3/3. Squared: (√3/3)^2 = 1/3z: √6/3. Squared: 6/9 = 2/3Sum: 1 + 1/3 + 2/3 = 1 + 1 = 2. So sqrt(2), which is CD=√2. Correct.So all distances check out. Therefore, the coordinates are correctly assigned.Therefore, the angle between AD and BC is indeed 60 degrees. So the answer is 60°, which in boxed form is boxed{60^circ} or boxed{dfrac{pi}{3}} radians. Since the problem doesn't specify, but in Olympiad problems, degrees are common unless stated otherwise. Let me check the original problem again. It says "the angle between AD and BC is qquad". It doesn't specify, but given the ratios are in sqrt(3), sqrt(2), which are common in 30-60-90 triangles, 45-45-90 triangles, so 60 degrees is likely the expected answer. So I think boxed{60^circ} is appropriate.Final Answerboxed{60^circ}</think>Given a tetrahedron (ABCD) with the edge ratios (BD: CD: AB: AC: AD: BC = sqrt{3}: sqrt{2}: 1: 1: 1: 1), we need to find the angle between edges (AD) and (BC).1. Assigning Coordinates: - Place (A) at the origin ((0, 0, 0)). - Place (B) at ((1, 0, 0)) since (AB = 1). - Place (C) at ((0.5, sqrt{3}/2, 0)) to form an equilateral triangle (ABC) with sides (AB = AC = BC = 1).2. Determining Coordinates of (D): - Let (D) have coordinates ((x, y, z)). - Using the given distances: - (AD = 1): (x^2 + y^2 + z^2 = 1) - (BD = sqrt{3}): ((x - 1)^2 + y^2 + z^2 = 3) - (CD = sqrt{2}): ((x - 0.5)^2 + (y - sqrt{3}/2)^2 + z^2 = 2) - Solving these equations, we find (x = -0.5), (y = sqrt{3}/6), and (z = sqrt{6}/3) (choosing the positive root for simplicity).3. Vectors Calculation: - Vector (AD) from (A(0, 0, 0)) to (D(-0.5, sqrt{3}/6, sqrt{6}/3)) is ((-0.5, sqrt{3}/6, sqrt{6}/3)). - Vector (BC) from (B(1, 0, 0)) to (C(0.5, sqrt{3}/2, 0)) is ((-0.5, sqrt{3}/2, 0)).4. Dot Product and Magnitudes: - Dot product of (AD) and (BC): [ (-0.5)(-0.5) + (sqrt{3}/6)(sqrt{3}/2) + (sqrt{6}/3)(0) = 0.25 + 0.25 + 0 = 0.5 ] - Magnitudes of (AD) and (BC) are both 1.5. Angle Calculation: - Using the dot product formula: [ cos theta = frac{AD cdot BC}{|AD| |BC|} = frac{0.5}{1 cdot 1} = 0.5 ] - Therefore, (theta = arccos(0.5) = 60^circ).The angle between (AD) and (BC) is (boxed{60^circ}).]

🔑:5. 60^{circ}.As shown in Figure 3, let AB = AC = AD = BC = 1, and denote the midpoints of BD, AB, and CD as E, F, and G respectively.Figure 3From AC^2 + AD^2 = CD^2 and G being the midpoint of CD, we get AG = sqrt{AD^2 - DG^2} = frac{sqrt{2}}{2}.Similarly, BG = frac{sqrt{6}}{2}.Also, AB = 1, AF = BF, and using the median length theorem, we get GF = frac{sqrt{2AG^2 + 2BG^2 - AB^2}}{2} = frac{sqrt{3}}{2}.Given EF = frac{1}{2} AD = frac{1}{2} and EG = frac{1}{2} BC = frac{1}{2}, we know angle FEG = 120^{circ}, which means the angle between EF and EG is 60^{circ}. Therefore, the angle between AD and BC is 60^{circ}.

🧠:[<think>Okay, let's see. Gillian has 50 songs that are 3 minutes each and 50 songs that are 5 minutes each. We need to figure out the maximum number of songs she can play in 3 hours. The options are given from A to E. Hmm, first, I need to convert 3 hours into minutes because the song lengths are in minutes. So, 3 hours is 180 minutes. Right?Now, the goal is to maximize the number of songs. Since the 3-minute songs are shorter, playing more of them would allow her to fit more songs into the 180 minutes. But she only has 50 of each. So, if she plays all 50 of the 3-minute songs, that would take 50 * 3 = 150 minutes. Then, she would have 180 - 150 = 30 minutes left. In those 30 minutes, how many 5-minute songs can she play? Well, 30 divided by 5 is 6. So she can play 6 of the 5-minute songs. That would be 50 + 6 = 56 songs total.But wait, let me check if that's right. 50 songs at 3 minutes is 150 minutes, then 6 songs at 5 minutes is 30 minutes. Total time is 150 + 30 = 180 minutes. Yep, that adds up. So that would give 56 songs. Hmm, option C is 56. But let me think if there's a way to get more songs. Maybe if she doesn't play all 50 of the 3-minute songs first?Wait, if she intersperses some 5-minute songs, maybe the leftover minutes can be used better? Let's see. Suppose she plays x number of 3-minute songs and y number of 5-minute songs. The total time would be 3x + 5y <= 180. We need to maximize x + y, given that x <= 50 and y <= 50.So, to maximize x + y, given 3x + 5y <= 180. Hmm, this is a linear programming problem. The maximum would occur at a vertex of the feasible region. The constraints are x <= 50, y <= 50, and 3x + 5y <= 180. Let's check the possible vertices.First, if we take x=50, then 3*50=150, so 5y <= 30 => y <= 6. So as before, x=50, y=6, total songs=56.Alternatively, if we take y=50, then 5*50=250, which is way more than 180. So y can't be 50. The maximum y possible is when x=0. Then 5y <=180 => y=36. So x=0, y=36, total songs=36. That's the option B, which is worse than 56.Another vertex would be where 3x + 5y = 180 and intersecting with another constraint. Maybe if x is less than 50? Let's suppose we don't use the maximum x or y. Let's solve 3x + 5y = 180. Let's express y in terms of x: y = (180 - 3x)/5. To have y as an integer, since you can't play a fraction of a song, (180 - 3x) must be divisible by 5. So 180 mod 5 is 0, and 3x mod 5 should be 0. Therefore, 3x ≡ 0 mod 5 => x ≡ 0 mod 5/ gcd(3,5)=1. So x ≡0 mod 5. So x must be a multiple of 5.Similarly, x can be 0,5,10,...,50. But x is limited to 50. So the possible maximum x is 50, which gives y=6 as before.Alternatively, if we consider that maybe using a combination where x is less than 50 and y is more than 6 but such that total songs x + y is more than 56. Let's see. Let's suppose x is 45, which is 45*3=135 minutes. Then 180 -135=45 minutes left. 45/5=9 songs. So y=9. Then total songs=45+9=54. That's less than 56. If x=40, 40*3=120, so 60 minutes left, y=12. Total songs=40+12=52. Still less. If x=55, but she only has 50 songs. So x cannot exceed 50.Alternatively, if x=50, y=6, total songs=56. If we try to see if we can get a higher total by using a different combination where x is slightly less than 50 and y is slightly more than 6, but such that x + y >56. For example, x=49. Then 3*49=147, 180-147=33. 33/5=6.6, so y=6.6, but we can't play a fraction of a song, so y=6, which gives total songs=49+6=55, which is less than 56. Similarly, x=48, 48*3=144, 180-144=36. 36/5=7.2, so y=7. Total songs=48+7=55. Still less than 56. Wait, 55? But 7 songs at 5 minutes would take 35 minutes, and 48*3=144. 144+35=179 minutes, leaving 1 minute unused. But you can't play a partial song, so yeah, that's 55 songs. Still less.How about x=55? No, she only has 50. So x can't go beyond 50. So maybe the maximum is indeed 56.Wait another angle: Suppose she plays a 5-minute song first. Let's say she plays 36 five-minute songs. 36*5=180. That would be 36 songs, which is option B, which is worse. So definitely, using more 3-minute songs is better.Alternatively, maybe she can play some combination where some of the 5-minute songs are replaced by more 3-minute songs. For example, replacing a 5-minute song with two 3-minute songs would save 5 - 2*3 = -1 minute. Wait, replacing a 5-minute song with two 3-minute songs would take 6 minutes instead of 5, which is worse. So that would actually take more time. So that's not helpful. Conversely, replacing a 3-minute song with a 5-minute song would save 3 -5 = -2 minutes, which also is worse. So perhaps there's no benefit in replacing.Alternatively, maybe she can play a combination where she uses the remaining time after 50 3-minute songs and 6 5-minute songs. But 50*3 +6*5=150+30=180, which exactly uses up the time. So there is no leftover time.Therefore, 56 is the maximum. So the answer should be C)56.Wait, but let me check if 56 is one of the options. Yes, option C is 56. So that must be the correct answer. The other options: A)100 is impossible because 100 songs even at 3 minutes would be 300 minutes, which is way over 180. B)36 is if she plays only 5-minute songs, which is worse. D)60 would require 60 songs. Let's see, if she plays 60 songs, if all 3-minute, that's 180 minutes. But she only has 50 songs that are 3 minutes. So she can play 50 3-minute songs (150 minutes) and 10 5-minute songs (50 minutes), but 150+50=200 minutes, which is over 180. So that doesn't work. Alternatively, maybe 55 3-minute and 5 5-minute: 55*3 +5*5=165 +25=190, still over. So 60 is not possible. E)80 is even worse. So yeah, 56 is the maximum. So the answer is C.Final Answerboxed{C}</think>Gillian has 50 songs that are 3 minutes each and 50 songs that are 5 minutes each. To determine the maximum number of songs she can play in 3 hours (180 minutes), we need to maximize the number of songs while considering the time constraints.First, converting 3 hours to minutes gives us 180 minutes. To maximize the number of songs, we should prioritize the shorter 3-minute songs. If Gillian plays all 50 of her 3-minute songs:- Time used: (50 times 3 = 150) minutes- Remaining time: (180 - 150 = 30) minutesIn the remaining 30 minutes, she can play:- Number of 5-minute songs: ( leftlfloor frac{30}{5} rightrfloor = 6 )Thus, the total number of songs played is:- (50 + 6 = 56)Checking other combinations, such as interspersing 3-minute and 5-minute songs, results in fewer total songs due to the constraints of the problem. For example, reducing the number of 3-minute songs and increasing the number of 5-minute songs leads to a lower total count.Therefore, the maximum number of songs Gillian can play in 3 hours is (boxed{C}).,<think>Okay, let's see. The problem is about Gillian wanting to play as many songs as possible within 3 hours. She has two types of songs: 50 songs that are 3 minutes each and 50 songs that are 5 minutes each. The question is asking for the maximum number of songs she can play in 3 hours. The options are A) 100, B) 36, C) 56, D) 60, and E) 80.First, I need to convert 3 hours into minutes because the song lengths are given in minutes. There are 60 minutes in an hour, so 3 hours would be 3 * 60 = 180 minutes. That's the total time she has to play songs.The goal is to maximize the number of songs. Since shorter songs take less time, playing more 3-minute songs would allow her to fit more songs into the 180 minutes. However, she only has 50 of the 3-minute songs. If she plays all 50, that would take 50 * 3 = 150 minutes. Then she would have 180 - 150 = 30 minutes left. In those 30 minutes, she could play 30 / 5 = 6 of the 5-minute songs. So that would be 50 + 6 = 56 songs total. Hmm, that's one of the options, option C.But wait, maybe there's a different combination that allows more songs? Let's check. If she plays some 5-minute songs along with the 3-minute ones, maybe the total number could be higher. But since 5-minute songs take longer, replacing a 5-minute song with a 3-minute song would save 2 minutes, which could allow another song. But she has a limited number of 3-minute songs (50). Let's see.Suppose she plays x 3-minute songs and y 5-minute songs. The total time is 3x + 5y ≤ 180. She wants to maximize x + y. The constraints are x ≤ 50 and y ≤ 50.To maximize x + y, we need to use as many short songs as possible. So first fill the time with as many 3-minute songs as possible. Since she can only play up to 50, that's 50 songs taking 150 minutes. Then use the remaining 30 minutes for 5-minute songs: 30 / 5 = 6. So 50 + 6 = 56.Alternatively, what if she plays fewer 3-minute songs to allow more 5-minute songs? Let's say she plays 49 3-minute songs. That would take 49 * 3 = 147 minutes. Remaining time is 180 - 147 = 33 minutes. 33 / 5 = 6.6, so 6 more 5-minute songs, which take 30 minutes, leaving 3 minutes. But she can't play another 5-minute song. So total songs would be 49 + 6 = 55, which is less than 56. So that's worse.Another way: Suppose she plays 48 3-minute songs. 48 * 3 = 144 minutes. Remaining 36 minutes. 36 / 5 = 7.2, so 7 songs. 7 * 5 = 35 minutes. Total time used: 144 + 35 = 179 minutes. Total songs: 48 + 7 = 55. Still less than 56.Alternatively, maybe use even fewer 3-minute songs. Let's try 45. 45 * 3 = 135 minutes. Remaining 45 minutes. 45 /5 = 9. So 45 + 9 = 54. Still lower.Alternatively, let's see if there's a way to use the remaining time better. For example, when she plays 50 3-minute songs, she uses 150 minutes, leaving 30 minutes. 30 minutes is exactly 6 5-minute songs, so that's perfect. So 50 + 6 = 56.If she tries to play 51 3-minute songs, but she only has 50. So that's not possible. So the maximum is 50 3-minute and 6 5-minute, totaling 56.Wait, but let's check another approach. What if she doesn't use all 50 3-minute songs? Suppose she uses 40 3-minute songs. Then time used: 40*3=120, remaining 60 minutes. Then 60 /5=12. So 40+12=52. That's worse than 56.Alternatively, maybe a different combination. For example, sometimes mixing the songs so that some leftover minutes can be used. But since 3 and 5 are coprime (their greatest common divisor is 1), in order to fill the time completely, it might not be possible. But since we want maximum number of songs, not necessarily filling the time, but fitting as many as possible.Alternatively, think of it as a linear programming problem. Let x be number of 3-minute songs, y the number of 5-minute songs. Maximize x + y, with constraints 3x +5y ≤180, x ≤50, y ≤50, x≥0, y≥0.To maximize x + y, given that 3x +5y ≤180. The optimal solution would be to have as much x as possible. So set x=50, then y=(180 - 3*50)/5 = (180 -150)/5=30/5=6. So that's x=50, y=6, sum 56. If x is less than 50, then y can be higher, but the sum x + y may not be higher. For example, if x=49, y= (180 -147)/5=33/5=6.6, so y=6, total 55. If x=55 (but she only has 50), so it's impossible.Alternatively, if there were no limits on the number of songs, you could solve 3x +5y ≤180. To maximize x + y, the optimal solution would be where the time per song is minimized. Since 3-minute songs take less time per song, we maximize x. So x=180/3=60. But since she only has 50, x=50, then y=6. So total 56. That's the maximum.Therefore, the answer is C) 56.Final Answerboxed{C}</think>Gillian has 3 hours (180 minutes) to play as many songs as possible from her collection of 50 songs that are 3 minutes each and 50 songs that are 5 minutes each. To maximize the number of songs, she should play as many 3-minute songs as possible since they take less time. Playing all 50 of the 3-minute songs uses:[ 50 times 3 = 150 text{ minutes} ]This leaves:[ 180 - 150 = 30 text{ minutes} ]In the remaining 30 minutes, she can play:[ leftlfloor frac{30}{5} rightrfloor = 6 text{ songs} ]of 5 minutes each.Therefore, the total number of songs she can play is:[ 50 + 6 = 56 ]Checking other combinations of 3-minute and 5-minute songs results in fewer total songs. Thus, the maximum number of songs Gillian can play in 3 hours is:[boxed{C}]]

🔑:To maximize the number of songs used, Gillian should use as many of the shortest length songs as possible. (This is because she can always trade a longer song for a shorter song and shorten the total time used.)If Gillian uses all 50 songs of 3 minutes in length, this takes 150 minutes.There are 180-150=30 minutes left, so she can play an additional 30 div 5=6 songs that are 5 minutes in length.In total, she plays 50+6=56 songs.ANSwer: (C)

🧠:[<think>Okay, let's try to figure out this geometry problem. So we have a right triangle ABC, where BC is longer than AC. It's inscribed in a circle, which means the hypotenuse must be the diameter of the circle because in a right-angled triangle, the hypotenuse is the diameter of the circumscribed circle. That's a theorem I remember. So the hypotenuse AB is the diameter of the circle.Now, point D is on BC such that BD equals AC. The problem is asking for the angle CDM, where M is the midpoint of the arc ACB. Hmm, arc ACB. Since ABC is a right triangle inscribed in the circle, the arc ACB must be the major arc, right? Because the hypotenuse AB is the diameter, so the circle is divided into two arcs by points A and B. The minor arc AB and the major arc ACB. So the midpoint of arc ACB would be the point opposite to the right angle C, maybe? Wait, in a circle, the midpoint of an arc is equidistant from the endpoints. So if M is the midpoint of arc ACB, then it's 180 degrees from... Wait, the arc ACB is the major arc, so M is the point such that arc from A to C to B to M would be half the circle? Wait, no. Wait, the entire circumference is 360 degrees, so arc ACB is 270 degrees? No, that doesn't make sense. Wait, if ABC is a right triangle inscribed in a circle with AB as the diameter, then points A, B, and C are on the circle. The arc from A to B via C would be 180 degrees because AB is the diameter. Wait, no. Wait, in a circle, if you have a diameter AB, then any point C on the circle forms a right angle at C. So the arc ACB would actually pass through C, which is on the circle. So the arc ACB is actually the semicircle from A to B through C, which is 180 degrees. But then the midpoint of that arc would be the point directly opposite to the midpoint of AB. Wait, but the midpoint of arc AB (the semicircle) is the point opposite the midpoint of AB. Wait, maybe not. Let me think again.If AB is the diameter, then the center of the circle is the midpoint of AB. The arc ACB is the semicircle containing point C. The midpoint of that arc would be the point M such that it's 90 degrees from both A and B along that semicircle. Wait, but if AB is the diameter, then the semicircle from A to B through C is 180 degrees. The midpoint of that arc would be the point where the angle subtended from AB is 90 degrees. Wait, maybe M is the point where if you draw a line from the center to M, it's perpendicular to AB? No, because the center is the midpoint of AB, so a line from the center perpendicular to AB would be the same as the altitude from the center, but AB is horizontal, say, then the perpendicular would be vertical. But in that case, point M would be at the top of the circle, directly above the center. But point C is somewhere on the circle. Wait, maybe M is the point such that angle AMB is 90 degrees? Wait, no, because M is the midpoint of the arc, so the arc length from A to M to B would be equal to the arc length from B to M to A? Wait, perhaps I need to draw a diagram mentally.Let me try to visualize this. Let's place the right triangle ABC with right angle at C. Since it's inscribed in a circle with AB as the diameter, the center of the circle is the midpoint of AB. Let's assume coordinates for simplicity. Let me assign coordinates to the points. Let’s set point A at (-1, 0) and point B at (1, 0), so the center is at (0,0), and the circle has radius 1. Then point C is somewhere on the circle, forming a right triangle. Since BC is longer than AC, point C should be in the upper half of the circle, closer to point B. Let's say point C has coordinates (x, y) on the unit circle. Then AC is the distance from (-1,0) to (x,y), which is sqrt((x+1)^2 + y^2), and BC is the distance from (1,0) to (x,y), sqrt((x-1)^2 + y^2). Since BC > AC, then sqrt((x-1)^2 + y^2) > sqrt((x+1)^2 + y^2). Squaring both sides: (x -1)^2 + y^2 > (x +1)^2 + y^2. Simplify: x^2 - 2x +1 + y^2 > x^2 + 2x +1 + y^2. Subtract x^2, y^2, 1 from both sides: -2x > 2x => -4x > 0 => x < 0. So point C is on the left side of the circle. Wait, but that contradicts BC > AC? Wait, if point C is on the left side, then BC would be longer? Wait, if point C is on the left, closer to A, then BC would be the distance from B (1,0) to C (-something, y), which could be longer. Wait, let's take an example: point C at (0,1). Then AC = sqrt((0+1)^2 +1^2) = sqrt(2), BC = sqrt((0-1)^2 +1^2) = sqrt(2). So equal. If point C is (-0.5, sqrt(3)/2), which is 120 degrees from positive x-axis, then AC is distance from (-1,0) to (-0.5, sqrt(3)/2): sqrt(0.5^2 + (sqrt(3)/2)^2) = sqrt(0.25 + 0.75) = sqrt(1) = 1. BC is distance from (1,0) to (-0.5, sqrt(3)/2): sqrt(1.5^2 + (sqrt(3)/2)^2) = sqrt(2.25 + 0.75) = sqrt(3) ≈1.732, which is greater than AC=1. So in this case, point C is on the left side (x=-0.5) but BC is longer. So indeed, if BC > AC, point C is on the left side of the circle (x negative). Okay, so coordinates: A(-1,0), B(1,0), C(-0.5, sqrt(3)/2), for example.Now, point D is on BC such that BD = AC. In our example, AC is 1, so BD should be 1. Since BC is sqrt(3), which is approximately 1.732, then D divides BC such that BD = 1 and DC = sqrt(3) -1 ≈0.732. So coordinates of D can be found by moving from B towards C by a distance of 1. Let's compute that.First, parametrize BC. From point B(1,0) to point C(-0.5, sqrt(3)/2). The vector from B to C is (-1.5, sqrt(3)/2). The length of BC is sqrt( (-1.5)^2 + (sqrt(3)/2)^2 ) = sqrt(2.25 + 0.75) = sqrt(3) ≈1.732. So to find point D which is 1 unit away from B towards C, we can parameterize the line BC.Parametric equations: x = 1 + t*(-1.5), y = 0 + t*(sqrt(3)/2), where t ranges from 0 to 1. The length from B to D is 1, which is t times the total length BC (sqrt(3)). So t = 1 / sqrt(3). Therefore, coordinates of D are:x = 1 - (1.5)*(1/sqrt(3)) = 1 - (3/(2 sqrt(3))) = 1 - (sqrt(3)/2) ≈1 -0.866≈0.134y = 0 + (sqrt(3)/2)*(1/sqrt(3)) = (sqrt(3)/2)*(1/sqrt(3)) = 1/2 ≈0.5So D is at approximately (0.134, 0.5). Wait, but in our coordinate system, point C is at (-0.5, sqrt(3)/2) ≈(-0.5, 0.866). So D is somewhere along BC, closer to B. But in coordinates, it's (1 - sqrt(3)/2, 0.5). Wait, sqrt(3)/2 ≈0.866, so 1 -0.866≈0.134. Yeah, so (0.134, 0.5).Now, point M is the midpoint of arc ACB. Since arc ACB is the semicircle from A to B through C, which is 180 degrees. The midpoint of this arc would be the point diametrically opposite to the center of the circle? Wait, no. The midpoint of arc ACB is the point that is equidistant from A and B along the arc. Wait, since AB is the diameter, the arc ACB is a semicircle. The midpoint of a semicircle is the point that is 90 degrees from both A and B. Wait, in our coordinate system, if AB is the diameter along the x-axis from (-1,0) to (1,0), then the midpoint of the arc ACB (the semicircle above the x-axis) is at (0,1). Wait, but point C is at (-0.5, sqrt(3)/2) ≈(-0.5, 0.866). But the midpoint of the arc ACB should be the highest point on the circle, which is (0,1). Because moving from A(-1,0) through C(-0.5, sqrt(3)/2) to B(1,0), the midpoint of that arc would be at the top of the circle. Wait, but let's check.The arc ACB is 180 degrees. The midpoint would be 90 degrees from both A and B. Starting at A(-1,0), moving 90 degrees along the circumference towards B(1,0) would reach (0,1). Similarly, starting at B(1,0), moving 90 degrees towards A(-1,0) would also reach (0,1). So yes, point M is at (0,1).So coordinates: M(0,1). Now, we need to find angle CDM. Points C, D, M.Point C is at (-0.5, sqrt(3)/2), D is at (1 - sqrt(3)/2, 0.5), and M is at (0,1). Let's compute vectors DC and DM, then find the angle between them.First, compute coordinates:C: (-0.5, sqrt(3)/2) ≈(-0.5, 0.866)D: (1 - sqrt(3)/2, 0.5) ≈(0.134, 0.5)M: (0,1)Vector DC is C - D: (-0.5 - (1 - sqrt(3)/2), sqrt(3)/2 - 0.5)Simplify:x-coordinate: -0.5 -1 + sqrt(3)/2 = -1.5 + sqrt(3)/2 ≈-1.5 +0.866≈-0.634y-coordinate: sqrt(3)/2 - 0.5 ≈0.866 -0.5≈0.366Vector DC: (-1.5 + sqrt(3)/2, sqrt(3)/2 - 0.5)Vector DM is M - D: (0 - (1 - sqrt(3)/2), 1 - 0.5)Simplify:x-coordinate: -1 + sqrt(3)/2 ≈-1 +0.866≈-0.134y-coordinate: 0.5Vector DM: (-1 + sqrt(3)/2, 0.5)To find angle CDM, which is the angle between vectors DC and DM at point D.The formula for the angle between two vectors u and v is:cosθ = (u • v) / (|u| |v|)Compute the dot product of DC and DM:u • v = [(-1.5 + sqrt(3)/2)(-1 + sqrt(3)/2)] + [(sqrt(3)/2 - 0.5)(0.5)]First compute the x-component:(-1.5 + sqrt(3)/2)(-1 + sqrt(3)/2)Let me compute this term by term:Multiply -1.5 by -1: 1.5Multiply -1.5 by sqrt(3)/2: -1.5*(sqrt(3)/2) ≈-1.5*0.866≈-1.299Multiply sqrt(3)/2 by -1: -sqrt(3)/2 ≈-0.866Multiply sqrt(3)/2 by sqrt(3)/2: (3)/4 = 0.75So total x-component:1.5 -1.299 -0.866 +0.75 ≈1.5 -1.299≈0.201; 0.201 -0.866≈-0.665; -0.665 +0.75≈0.085Now y-component:(sqrt(3)/2 - 0.5)(0.5) = (sqrt(3)/2 *0.5) - (0.5 *0.5) = sqrt(3)/4 -0.25 ≈0.433 -0.25≈0.183Total dot product: 0.085 +0.183≈0.268Now compute |u| (magnitude of DC):sqrt[ (-1.5 + sqrt(3)/2)^2 + (sqrt(3)/2 -0.5)^2 ]First term: (-1.5 + sqrt(3)/2)^2 ≈(-1.5 +0.866)^2≈(-0.634)^2≈0.402Second term: (sqrt(3)/2 -0.5)^2≈(0.866 -0.5)^2≈(0.366)^2≈0.134Sum: 0.402 +0.134≈0.536; sqrt(0.536)≈0.732|u|≈0.732|v| (magnitude of DM):sqrt[ (-1 + sqrt(3)/2)^2 + (0.5)^2 ]First term: (-1 +0.866)^2≈(-0.134)^2≈0.018Second term: 0.25Sum: 0.018 +0.25≈0.268; sqrt(0.268)≈0.517|v|≈0.517So cosθ≈0.268 / (0.732 *0.517)≈0.268 /0.379≈0.707So θ≈arccos(0.707)≈45 degrees.Wait, that's interesting. So angle CDM is 45 degrees? Let me check if this is exact.Wait, cosθ≈0.707 is approximately sqrt(2)/2≈0.7071, which corresponds to 45 degrees. So in this coordinate system, the angle is exactly 45 degrees?Wait, maybe in the specific case I chose, but is this always true regardless of the position of point C? Because I chose a specific coordinate system where AB is the diameter of the unit circle. But the problem didn't specify coordinates, so maybe this is a general result.Alternatively, maybe there's a geometric theorem or property that makes this angle 45 degrees regardless of the triangle's dimensions, as long as BD = AC and M is the midpoint of the arc.Let me see if I can find a synthetic solution without coordinates.Given triangle ABC is right-angled at C, inscribed in a circle with AB as diameter. M is the midpoint of arc ACB, so M is the point diametrically opposite to the center? Wait, no. Wait, since AB is the diameter, the center is the midpoint of AB. The arc ACB is the semicircle not containing the opposite side? Wait, no, all points on the circle are part of the semicircle with AB as diameter.Wait, actually, in a circle, any right-angled triangle inscribed in the circle must have its right angle on the circle with the hypotenuse as diameter. So ABC is right-angled at C, so AB is the diameter. The arc ACB is the semicircle from A to B passing through C. The midpoint of this arc is the point M such that AM = BM, and M is the highest point on the circle, directly above the center.So in general, M is the point where the angle AMB is 90 degrees? Wait, no. If M is the midpoint of arc AB (the semicircle), then angles from M to AB would be right angles. Wait, maybe angle AMB is 90 degrees? Wait, since M is on the circle, by Thales' theorem, angle AMB would be 90 degrees if AB is the diameter. But AB is the diameter, so angle AMB is 90 degrees. Wait, but point M is on the circle, so angle AMB is 90 degrees. But M is also the midpoint of arc ACB. Wait, but in our coordinate system, M is (0,1), A is (-1,0), B is (1,0), so triangle AMB is a right triangle at M. So angle at M is 90 degrees.Wait, but how does this help us find angle CDM?Alternatively, maybe we can use some properties of circles and midpoints. Let's consider that BD = AC. Since ABC is a right triangle, AC and BC are the legs, with BC > AC. BD is a segment on BC equal to AC.Perhaps triangles ABC and BDM have some relation? Let's see.Wait, BD = AC, and maybe angle at B is common? Let's check.In triangle ABC, angle at B is angle ABC. In triangle DBM, angle at B is the same. If BD = AC and perhaps some sides are proportional, maybe the triangles are similar? But ABC is a right triangle, and DBM is not necessarily.Alternatively, since M is the midpoint of the arc ACB, it might lie on the angle bisector or perpendicular bisector of some segment.Alternatively, since M is the midpoint of the arc ACB, the line from M to the center (midpoint of AB) is perpendicular to AB. In our coordinate system, that's the y-axis. So line MC might have some symmetry.Alternatively, since M is the midpoint of arc ACB, then MA = MB = MC in some sense? Wait, no. In a circle, all points on the circle are equidistant from the center, but arcs correspond to angles. Since M is the midpoint of arc ACB, then angles from M to A and M to B are equal. So angle MAC = angle MBC? Wait, maybe not. Wait, the arcs from M to A and M to B would each be 90 degrees, since the total arc ACB is 180 degrees, so each part is 90 degrees. Therefore, angles subtended by those arcs would be 45 degrees? Wait, maybe.Alternatively, since M is the midpoint of arc ACB, then the line from M to the center is the perpendicular bisector of AB. So in coordinates, it's the y-axis. So point M is (0,1). Then line DM is from D(1 - sqrt(3)/2, 0.5) to M(0,1). Let me see if there's a reflection or symmetry here.Alternatively, since BD = AC, perhaps there is a reflection or rotation that maps AC to BD. Let me consider triangle ABC. AC is one leg, BD is a segment on BC equal to AC. Maybe triangle ACD is congruent or similar to some other triangle.Alternatively, consider drawing line DM and trying to find some cyclic quadrilateral or something.Wait, maybe using power of a point. Point D with respect to the circle. But D is inside the circle, so power of D would be negative, equal to the product of distances along a secant line. But not sure.Alternatively, consider coordinates again. In my coordinate system, angle CDM came out to be 45 degrees. Let me check if this is exact.Wait, in the coordinate system where AB is the diameter of the unit circle, with A(-1,0), B(1,0), C at some point (-x, y) on the circle (x >0). Then BD = AC.First, compute AC: distance between A(-1,0) and C(-x,y) is sqrt( (-x +1)^2 + y^2 )Since C is on the circle, x^2 + y^2 =1.BD is the length from B(1,0) to D on BC such that BD = AC. Let's parametrize D.Parametric coordinates of D: moving from B(1,0) towards C(-x,y), a fraction t of the total distance BC.Coordinates of D: (1 - t(1 +x), t y )Distance BD: t * BC. Since BD = AC, then t = AC / BC.But AC = sqrt( (1 -x)^2 + y^2 )But since x^2 + y^2 =1, (1 -x)^2 + y^2 = 1 - 2x +x^2 + y^2 = 2 - 2x.Therefore, AC = sqrt(2 - 2x)BC is the distance from B(1,0) to C(-x,y): sqrt( (1 +x)^2 + y^2 ) = sqrt(1 + 2x +x^2 + y^2 ) = sqrt(2 + 2x )Because x^2 + y^2 =1, so 1 + 2x +x^2 + y^2 =1 +2x +1=2 +2x.Therefore, BC = sqrt(2 + 2x )Therefore, t = AC / BC = sqrt(2 - 2x ) / sqrt(2 + 2x ) = sqrt( (2 -2x)/(2 +2x) ) = sqrt( (1 -x)/(1 +x) )Therefore, coordinates of D:x-coordinate: 1 - t(1 +x ) =1 - sqrt( (1 -x)/(1 +x) )*(1 +x ) =1 - sqrt( (1 -x)(1 +x) ) =1 - sqrt(1 -x^2 )Similarly, y-coordinate: t y = sqrt( (1 -x)/(1 +x) ) * yBut since x^2 + y^2 =1, y = sqrt(1 -x^2 )Therefore, y-coordinate of D: sqrt( (1 -x)/(1 +x) ) * sqrt(1 -x^2 ) = sqrt( (1 -x)/(1 +x) ) * sqrt( (1 -x)(1 +x) ) )= sqrt( (1 -x)^2 ) =1 -xWait, that's interesting. So the y-coordinate of D is 1 -x.Therefore, coordinates of D are (1 - sqrt(1 -x^2 ), 1 -x )Point M is the midpoint of arc ACB, which is (0,1) as we saw earlier.Now, we need to find angle CDM. Points C(-x, sqrt(1 -x^2 )), D(1 - sqrt(1 -x^2 ),1 -x ), M(0,1 )Compute vectors DC and DM.Vector DC: C - D = (-x - (1 - sqrt(1 -x^2 )), sqrt(1 -x^2 ) - (1 -x ) )Simplify x-component: -x -1 + sqrt(1 -x^2 )y-component: sqrt(1 -x^2 ) -1 +xVector DM: M - D = (0 - (1 - sqrt(1 -x^2 )),1 - (1 -x )) = ( -1 + sqrt(1 -x^2 ), x )So vectors:DC: ( -1 -x + sqrt(1 -x^2 ), x -1 + sqrt(1 -x^2 ) )DM: ( -1 + sqrt(1 -x^2 ), x )Now, compute the dot product of DC and DM:[ (-1 -x + sqrt(1 -x^2 )) * (-1 + sqrt(1 -x^2 )) ] + [ (x -1 + sqrt(1 -x^2 )) * x ]Let's compute the first term:Let me denote sqrt(1 -x^2 ) as S for simplicity.First term: (-1 -x + S)(-1 + S ) = (-1)(-1) + (-1)(S) + (-x)(-1) + (-x)(S) + S*(-1) + S*SWait, perhaps expand as:= (-1 -x + S)(-1 + S )Multiply term by term:First, multiply -1 by (-1 + S ): 1 - SThen multiply -x by (-1 + S ): x - x SThen multiply S by (-1 + S ): -S + S^2Add them all up:1 - S + x - x S - S + S^2Combine like terms:1 + x -2S -x S + S^2Second term: (x -1 + S ) *x = x^2 -x +x SSo total dot product:[1 + x -2S -x S + S^2 ] + [x^2 -x +x S ] =1 + x -2S -x S + S^2 +x^2 -x +x SSimplify:1 + (x -x) + (-2S) + (-x S +x S ) + S^2 +x^2Which simplifies to:1 -2S + S^2 +x^2Now, note that S = sqrt(1 -x^2 ), so S^2 =1 -x^2Therefore:1 -2S + (1 -x^2 ) +x^2 =1 -2S +1 -x^2 +x^2 =2 -2SSo dot product =2(1 - S )Now, compute the magnitudes of DC and DM.First, |DC|:sqrt( ( -1 -x + S )^2 + (x -1 + S )^2 )Let me compute each component squared:First component squared: (-1 -x + S )^2 = ( - (1 +x ) + S )^2 = (S - (1 +x ))^2 = S^2 - 2S(1 +x ) + (1 +x )^2Second component squared: (x -1 + S )^2 = (S +x -1 )^2 = S^2 + 2S(x -1 ) + (x -1 )^2Add them together:S^2 -2S(1 +x ) + (1 +x )^2 + S^2 + 2S(x -1 ) + (x -1 )^2Combine like terms:2S^2 -2S(1 +x ) +2S(x -1 ) + (1 +x )^2 + (x -1 )^2Simplify the S terms:-2S(1 +x ) +2S(x -1 ) = -2S -2x S +2x S -2S = -4SThe other terms:(1 +x )^2 + (x -1 )^2 = (1 +2x +x^2 ) + (1 -2x +x^2 )= 2 +2x^2Therefore, |DC|^2 =2S^2 -4S +2 +2x^2But S^2 =1 -x^2, so:2(1 -x^2 ) -4S +2 +2x^2 =2 -2x^2 -4S +2 +2x^2 =4 -4STherefore, |DC|=sqrt(4 -4S )=2*sqrt(1 -S )Similarly, compute |DM|:sqrt( (-1 + S )^2 +x^2 )= sqrt(1 -2S +S^2 +x^2 )Again, S^2 =1 -x^2, so:1 -2S +1 -x^2 +x^2 =2 -2STherefore, |DM|=sqrt(2 -2S )=sqrt(2(1 -S ))=sqrt(2)*sqrt(1 -S )Therefore, the dot product is 2(1 -S ), and |DC||DM|=2*sqrt(1 -S )*sqrt(2)*sqrt(1 -S )=2*sqrt(2)*(1 -S )Therefore, cosθ= [2(1 -S )]/[2*sqrt(2)*(1 -S ) ]=1/sqrt(2 )Thus, θ=45 degrees.Wow! So regardless of the value of x (as long as BC > AC, which implies x <0 in our coordinate system), the angle CDM is always 45 degrees. Therefore, the answer is 45 degrees, which is π/4 radians.So, the angle CDM is 45 degrees. Therefore, the answer is boxed{45^circ}Final Answerboxed{45^circ}</think>Given a right triangle (ABC) with (BC > AC) inscribed in a circle (with (AB) as the diameter), we need to find the angle (angle CDM) where (D) is a point on (BC) such that (BD = AC) and (M) is the midpoint of the arc (ACB).1. Circle and Midpoint Properties: - Since (ABC) is a right triangle inscribed in a circle, the hypotenuse (AB) is the diameter of the circle. - The midpoint (M) of the arc (ACB) (the semicircle containing (C)) is the point diametrically opposite the midpoint of (AB), which is the highest point on the circle ((0, 1)) in a coordinate system where (A) and (B) are at ((-1, 0)) and ((1, 0)) respectively.2. Coordinates and Vectors: - Assign coordinates: (A(-1, 0)), (B(1, 0)), and (C) on the circle. For simplicity, consider (C) at ((-x, y)) where (x^2 + y^2 = 1). - Point (D) is on (BC) such that (BD = AC). Using parametric coordinates and vector analysis, we find the coordinates of (D) as ((1 - sqrt{1 - x^2}, 1 - x)).3. Vectors and Dot Product: - Vectors ( overrightarrow{DC} ) and ( overrightarrow{DM} ) are computed. - The dot product and magnitudes of these vectors are calculated, leading to the cosine of the angle between them.4. Angle Calculation: - Using the dot product formula, we find that the cosine of the angle (angle CDM) is (frac{1}{sqrt{2}}), which corresponds to (45^circ).Thus, the angle (angle CDM) is (boxed{45^circ}).,<think>Okay, let's see. So, we have a right triangle ABC where BC is longer than AC. It's inscribed in a circle, which makes sense because a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. That's a property I remember from geometry. So, the hypotenuse AB is the diameter of the circle. Therefore, the center of the circle must be the midpoint of AB. Wait, but the problem mentions point M is the midpoint of the arc ACB. Hmm, since ABC is a right triangle with right angle at C, the hypotenuse AB is the diameter, so the arc ACB must be the major arc passing through points A, C, and B. The midpoint of that arc would be the point opposite to the right angle C, right? Because in a circle, the midpoint of an arc is the point where the angle subtended by the arc is half the measure of the arc. Since AB is the diameter, the arc AB is 180 degrees, but the major arc ACB would be 180 degrees plus the arc CB. Wait, maybe I need to think more carefully.Wait, actually, since ABC is a right triangle, the circle in which it's inscribed has AB as the diameter. So, points A, B, and C lie on the circle with AB as diameter. The arc ACB is the major arc that goes from A to C to B, which is 180 degrees plus the angle at C. But since ABC is a right triangle, angle at C is 90 degrees, so the arc ACB would be 270 degrees? Wait, no. Wait, arcs on a circle are measured by the central angles. Since AB is the diameter, the central angle over AB is 180 degrees. The point C is somewhere on the circle such that ACB is a right angle. So, the arc from A to B not containing C is 180 degrees, and the arc ACB would actually be the other half of the circle? Wait, that doesn't make sense. Wait, maybe the arc ACB is 180 degrees as well. Wait, no.Let me recall that in a circle, the measure of an arc is twice the measure of the inscribed angle that subtends it. Since angle at C is 90 degrees, the arc opposite to it, which is arc AB, must be 180 degrees. So, arc AB is 180 degrees. Then, the rest of the circle is another 180 degrees, which would be the arc ACB. But wait, arc ACB would actually pass through C, so maybe the arc ACB is 270 degrees? Wait, no. Wait, if AB is the diameter, then the circle is divided into two semicircles by AB. The point C is on one of those semicircles. If ABC is a right triangle, then C must lie on the circle with AB as diameter. So, the arc from A to B through C is 180 degrees. Then, the other arc from A to B not through C is also 180 degrees. Wait, but the problem says M is the midpoint of the arc ACB. So, arc ACB is the major arc, meaning it's the longer arc from A to B passing through C. But since AB is the diameter, both arcs from A to B are 180 degrees each. That can't be. Therefore, maybe the triangle is not necessarily in a semicircle? Wait, but a right triangle inscribed in a circle must have its hypotenuse as the diameter. So, AB must be the diameter, so the arc ACB is 180 degrees. Then, the midpoint of arc ACB would be the point opposite to C? Wait, but if the arc is 180 degrees, then its midpoint would be the point diametrically opposite to the midpoint of AB? Wait, maybe I need to draw a diagram mentally.Let me try to visualize this. Let's have triangle ABC with right angle at C. AB is the hypotenuse and the diameter of the circle. The circle has center at the midpoint of AB. The arc ACB is the semicircle from A to B passing through C. The midpoint of that arc would be the point opposite to the midpoint of AB? Wait, no. The midpoint of a semicircular arc would be the point that's 90 degrees from both A and B along the arc. Wait, but in this case, since AB is the diameter, the arc ACB is a semicircle. The midpoint of that semicircle would be the point that's 90 degrees away from both A and B along the circumference. But since AB is a straight line, the midpoint of the arc ACB would be the point where the perpendicular bisector of AB meets the circle. But the perpendicular bisector of AB is the line that passes through the center of the circle (since AB is the diameter), so it's the same as the center. Wait, no. The perpendicular bisector of AB is the line that's at 90 degrees to AB at its midpoint, which is the center. But if we're talking about the midpoint of the arc ACB, which is a semicircle, then the midpoint would be the point on the arc that is equidistant from A and B along the circumference. That point is the same as the point where the altitude from C to AB meets the circle? Wait, no.Wait, perhaps it's simpler. In a semicircle with diameter AB, any point C on the semicircle forms a right triangle ABC. The arc ACB is the semicircle itself. The midpoint of that arc would be the point M such that it's halfway between A and B along the semicircle. But since the semicircle is 180 degrees, the midpoint would be 90 degrees from both A and B. But since AB is the diameter, moving 90 degrees from A along the semicircle would reach the point that's at the top of the circle, directly above the center. Similarly, moving 90 degrees from B would reach the same point. Wait, that seems contradictory. Wait, if you start at A and move 90 degrees along the semicircle (which is 180 degrees total), then the midpoint would be 90 degrees away from A. Similarly, 90 degrees from B. Therefore, point M is the point on the semicircle ACB such that arc AM and arc MB are each 90 degrees. But in this case, since AB is 180 degrees, each 90 degrees would be a quarter of the circle. Wait, but the entire circle is 360 degrees. If AB is a diameter, then the semicircle ACB is 180 degrees. The midpoint of that arc would divide it into two arcs of 90 degrees each. So, starting from A, moving 90 degrees along the arc ACB brings us to point M, and from M another 90 degrees to B. Therefore, point M is located such that angle AOM is 90 degrees, where O is the center. Since O is the midpoint of AB, and the radius is half of AB, then point M is the point perpendicular to AB above the center. Wait, yes. If you have a diameter AB horizontally, then the center O is the midpoint. The point M would be at the top of the circle, forming a right angle with AB. So, coordinate-wise, if AB is from (-r, 0) to (r, 0), then M would be at (0, r). But in our case, since ABC is a right triangle with right angle at C, point C is somewhere on the circle. Since BC > AC, then point C is closer to A than to B along the semicircle.Wait, maybe assigning coordinates would help. Let me try to set up a coordinate system. Let's let AB be the diameter along the x-axis, with the center at the origin O (0,0). Let’s let point A be (-1, 0) and point B be (1, 0). Then the circle has radius 1. The right triangle ABC has right angle at C, so point C is somewhere on the circle. Since BC > AC, which in terms of coordinates would mean that the length from B to C is greater than from A to C. So, point C is in the upper half of the circle (since the triangle is right-angled at C), and closer to point A than to point B. Therefore, its x-coordinate would be closer to -1 than to 1. Let's denote point C as (x, y), where x^2 + y^2 = 1 (since it's on the unit circle). Then, AC is the distance from (-1,0) to (x,y): sqrt((x+1)^2 + y^2). BC is the distance from (1,0) to (x,y): sqrt((x-1)^2 + y^2). The condition BC > AC implies sqrt((x-1)^2 + y^2) > sqrt((x+1)^2 + y^2). Squaring both sides: (x - 1)^2 + y^2 > (x + 1)^2 + y^2. Expanding: x^2 - 2x + 1 + y^2 > x^2 + 2x + 1 + y^2. Simplify: -2x > 2x => -4x > 0 => x < 0. So, point C has a negative x-coordinate, which makes sense since it's closer to A.So, let's assign coordinates. Let me set AB as the diameter from (-1,0) to (1,0), center at (0,0). Point C is (x, y) on the unit circle with x < 0. Then, point D is on BC such that BD = AC. Let's compute AC and BD. AC is the distance from A to C: sqrt((x + 1)^2 + y^2). Since C is on the unit circle, x^2 + y^2 = 1. Therefore, AC = sqrt((x + 1)^2 + (1 - x^2)) because y^2 = 1 - x^2. Let's compute that: (x + 1)^2 + (1 - x^2) = x^2 + 2x + 1 + 1 - x^2 = 2x + 2. Therefore, AC = sqrt(2x + 2) = sqrt(2(x + 1)). Hmm, interesting.BD is equal to AC, so BD = sqrt(2(x + 1)). But BD is a length on BC. Let's find coordinates of D. Since D is on BC, we can parametrize BC. Point B is (1, 0), point C is (x, y). Let's parametrize BC as follows: from B (1,0) to C (x,y). The vector from B to C is (x - 1, y - 0) = (x - 1, y). The length of BC is sqrt((x - 1)^2 + y^2) = sqrt((x - 1)^2 + (1 - x^2)) similar to AC. Let's compute that: (x - 1)^2 + (1 - x^2) = x^2 - 2x + 1 + 1 - x^2 = -2x + 2. Therefore, BC = sqrt(-2x + 2) = sqrt(2(1 - x)).We need BD = AC. Since BD is a part of BC, BD = sqrt(2(x + 1)) and BC = sqrt(2(1 - x)). So, BD / BC = sqrt(2(x + 1)) / sqrt(2(1 - x)) ) = sqrt( (x + 1)/(1 - x) ). Let's denote t = BD / BC = sqrt( (x + 1)/(1 - x) ). Then, t^2 = (x + 1)/(1 - x) => t^2(1 - x) = x + 1 => t^2 - t^2 x = x + 1 => -t^2 x - x = 1 - t^2 => x(-t^2 - 1) = 1 - t^2 => x = (1 - t^2)/(-t^2 - 1) = (t^2 - 1)/(t^2 + 1). Hmm, but I'm not sure if this is the right approach.Alternatively, since BD = AC, and we have coordinates for B, C, we can find the coordinates of D. Let's parameterize point D along BC. Let’s say D divides BC in the ratio k:1, where k = BD / DC. Since BD = AC, and AC = sqrt(2(x + 1)), and BC = sqrt(2(1 - x)), then BD / BC = sqrt( (x + 1)/(1 - x) ). Therefore, BD = sqrt( (x + 1)/(1 - x) ) * BC. Wait, maybe this is getting too complicated. Perhaps another approach.Alternatively, since D is on BC such that BD = AC, we can express D in coordinates. Let me denote coordinates of C as (x, y). Then, vector BC is (x - 1, y). Let's parameterize D as moving from B towards C by a distance equal to AC. Since BD = AC, which is sqrt(2(x + 1)). The unit vector in the direction from B to C is (x - 1, y) divided by BC, which is sqrt(2(1 - x)). Therefore, the coordinates of D would be B plus BD times the unit vector: D = (1, 0) + sqrt(2(x + 1)) * ( (x - 1)/sqrt(2(1 - x)) , y/sqrt(2(1 - x)) )Simplify the components:x-coordinate: 1 + sqrt(2(x + 1)) * (x - 1)/sqrt(2(1 - x)) ) = 1 + (x - 1) * sqrt( (x + 1)/(1 - x) )Similarly, y-coordinate: 0 + sqrt(2(x + 1)) * y / sqrt(2(1 - x)) ) = y * sqrt( (x + 1)/(1 - x) )Let me compute sqrt( (x + 1)/(1 - x) ). Let's denote that as t. Then t = sqrt( (x + 1)/(1 - x) )Therefore, x-coordinate of D is 1 + (x - 1) * t, and y-coordinate is y * t.But let's compute t^2 = (x + 1)/(1 - x) => t^2(1 - x) = x + 1 => t^2 - t^2 x = x + 1 => - t^2 x - x = 1 - t^2 => x(-t^2 - 1) = 1 - t^2 => x = (1 - t^2)/(-t^2 -1 ) = (t^2 -1)/(t^2 +1 )Hmm, not sure if this is helpful. Maybe instead of parameterizing, we can use coordinates for D.Alternatively, since BD = AC, and AC is sqrt(2(x +1)), perhaps we can express D as (1 - (sqrt(2(x +1))/BC)*(1 - x), 0 + (sqrt(2(x +1))/BC)*y ). Since BD is along BC, so the coordinates would be B plus the vector from B to C scaled by BD / BC.So, BD / BC = sqrt(2(x +1)) / sqrt(2(1 -x)) ) = sqrt( (x +1)/(1 -x) )Therefore, coordinates of D:x-coordinate: 1 + (x - 1) * sqrt( (x +1)/(1 -x) )y-coordinate: 0 + y * sqrt( (x +1)/(1 -x) )Simplify x-coordinate:1 + (x -1) * sqrt( (x +1)/(1 -x) ) = 1 - (1 - x) * sqrt( (x +1)/(1 -x) ) = 1 - sqrt( (1 - x)^2 * (x +1)/(1 -x) ) = 1 - sqrt( (1 - x)(x +1) )Which simplifies to 1 - sqrt(1 - x^2 )Similarly, y-coordinate:y * sqrt( (x +1)/(1 -x) )But since y = sqrt(1 - x^2), because C is on the unit circle. So, substituting y:sqrt(1 - x^2 ) * sqrt( (x +1)/(1 -x) ) = sqrt( (1 - x^2)(x +1)/(1 -x) ) = sqrt( ( (1 -x)(1 +x) )(x +1)/(1 -x) ) ) = sqrt( (1 +x)^2 ) = 1 + xTherefore, coordinates of D are:x-coordinate: 1 - sqrt(1 - x^2 )y-coordinate: 1 + xSo, D is (1 - sqrt(1 - x^2 ), 1 + x )Wait, that seems interesting. Let me verify that.Starting from:x-coordinate of D: 1 - sqrt(1 - x^2 )y-coordinate of D: 1 + xHmm, let's check with a specific example. Suppose point C is at (-1, 0). Wait, but that's point A. No, point C cannot be A because then BC would be BA, which is longer than AC=0. Wait, but in our setup, point C is on the circle with x < 0. Let's take point C at (0,1). Then x=0, y=1. Then AC is sqrt((0 +1)^2 +1^2 )=sqrt(2). BD should be equal to AC=sqrt(2). BC is the distance from (1,0) to (0,1): sqrt( (0 -1)^2 + (1 -0)^2 ) = sqrt(2). So BD = sqrt(2), which is equal to BC. Therefore, D would be point C. But according to the coordinates we derived:x-coordinate of D: 1 - sqrt(1 -0^2 )=1 -1=0y-coordinate:1 +0=1, which is point (0,1), which is C. Correct. So in this case, when AC=BC, which contradicts the condition BC > AC. Wait, but in this case, AC=sqrt(2), BC=sqrt(2), so BC is not greater than AC, but equal. But according to the problem, BC > AC, so point C cannot be at (0,1). Let's take another point. Suppose point C is at (-0.5, sqrt(1 - 0.25)) = (-0.5, sqrt(0.75)). Then AC is distance from (-1,0) to (-0.5, sqrt(0.75)): sqrt( (0.5)^2 + (sqrt(0.75))^2 ) = sqrt(0.25 + 0.75 )=sqrt(1)=1. BC is distance from (1,0) to (-0.5, sqrt(0.75)): sqrt( (1.5)^2 + (sqrt(0.75))^2 )=sqrt(2.25 + 0.75)=sqrt(3)≈1.732. So BD should be equal to AC=1. Then BD=1, so D is 1 unit away from B along BC. The total length BC is sqrt(3), so BD/BC=1/sqrt(3). Therefore, coordinates of D would be:x-coordinate: 1 + ( (-0.5 -1) )*(1/sqrt(3)) =1 + (-1.5)/sqrt(3)y-coordinate:0 + sqrt(0.75)*(1/sqrt(3))= sqrt(0.75)/sqrt(3)=sqrt(0.75/3)=sqrt(0.25)=0.5Wait, but according to our previous formula:x-coordinate of D:1 - sqrt(1 - x^2 ). x here is -0.5, so x^2=0.25. sqrt(1 - x^2)=sqrt(0.75). Therefore, x-coordinate of D:1 - sqrt(0.75)≈1 - 0.866≈0.134y-coordinate:1 + x=1 + (-0.5)=0.5So coordinates are approximately (0.134, 0.5). Let's check if this matches the previous calculation.From parametrization:D is 1/sqrt(3) of the way from B to C. So:x-coordinate:1 + (-1.5)*(1/sqrt(3))≈1 - 1.5/1.732≈1 -0.866≈0.134y-coordinate:0 + sqrt(0.75)*(1/sqrt(3))≈0.866*0.577≈0.5Yes, matches. So the formula seems correct.Therefore, in general, coordinates of D are (1 - sqrt(1 - x^2 ), 1 + x )But x is the x-coordinate of point C, which is between -1 and 0.Now, point M is the midpoint of arc ACB. Since arc ACB is the semicircle from A to B through C, its midpoint is the point diametrically opposite to the midpoint of AB. Wait, no. Wait, the midpoint of arc ACB is the point M such that it's 90 degrees from both A and B? Wait, but in the semicircle, moving 90 degrees from A would reach the top of the circle. Wait, but the arc ACB is 180 degrees. The midpoint would be the point that's 90 degrees away from both A and B along that arc. But since AB is the diameter, the midpoint of the arc ACB is the point at (0,1) if AB is along the x-axis from (-1,0) to (1,0). Wait, but if AB is the diameter, then the semicircle ACB is the upper half. Then, the midpoint of arc ACB would be (0,1). Wait, yes. Because moving 90 degrees from A (which is (-1,0)) along the semicircle brings you to (0,1), and similarly from B. So point M is (0,1).Wait, but in our coordinate system, point M is (0,1). Let me verify. If AB is from (-1,0) to (1,0), then the arc ACB is the upper semicircle. The midpoint of this arc is indeed (0,1). So regardless of where point C is on the semicircle, the midpoint M is fixed at (0,1). That simplifies things.Therefore, point M is (0,1). Now, we need to find angle CDM. That is, angle at point D between points C, D, and M.So, points:C: (x, y) = (x, sqrt(1 - x^2 )) with x < 0.D: (1 - sqrt(1 - x^2 ), 1 + x )M: (0,1)We need to compute angle CDM, which is the angle at D between points C, D, M.To find angle CDM, we can compute vectors DC and DM and find the angle between them.First, coordinates:Point C: (x, sqrt(1 - x^2 ))Point D: (1 - sqrt(1 - x^2 ), 1 + x )Point M: (0,1)Vector DC = C - D = (x - (1 - sqrt(1 - x^2 )), sqrt(1 - x^2 ) - (1 + x ) )Vector DM = M - D = (0 - (1 - sqrt(1 - x^2 )), 1 - (1 + x )) = ( sqrt(1 - x^2 ) - 1, -x )Compute angle between vectors DC and DM.The angle θ between vectors u and v is given by:cosθ = (u • v) / (|u| |v| )First, compute u = DC:u_x = x - 1 + sqrt(1 - x^2 )u_y = sqrt(1 - x^2 ) - 1 - xv = DM:v_x = sqrt(1 - x^2 ) - 1v_y = -xCompute the dot product u • v:= [x - 1 + sqrt(1 - x^2 )] * [sqrt(1 - x^2 ) - 1] + [sqrt(1 - x^2 ) - 1 - x] * (-x )Let me compute each part step by step.First term:[x - 1 + sqrt(1 - x^2 )] * [sqrt(1 - x^2 ) - 1]Let me denote sqrt(1 - x^2 ) as s for simplicity.Then, first term becomes [x -1 + s] * [s -1] = (x -1)(s -1) + s(s -1)= (x -1)s - (x -1) + s^2 - sSecond term:[sqrt(1 - x^2 ) -1 -x ] * (-x ) = [s -1 -x ]*(-x ) = -x(s -1 -x ) = -x s +x +x^2So total dot product:First term + second term:[(x -1)s - (x -1) + s^2 - s] + [ -x s +x +x^2 ]Let me expand:= (x -1)s - x +1 + s^2 - s -x s +x +x^2Simplify term by term:(x -1)s -x s = x s - s -x s = -s-s-s - s = -2sThen:- x +1 +x = 1+ s^2 +x^2So overall:-2s +1 + s^2 +x^2But s = sqrt(1 - x^2 ), so s^2 =1 -x^2. Therefore:-2s +1 + (1 -x^2 ) +x^2 = -2s +1 +1 -x^2 +x^2 = -2s +2Therefore, the dot product u • v = -2s +2 = 2(1 - s )Now, compute |u| and |v|.First, |u|:|u| = sqrt( (x -1 + s)^2 + (s -1 -x)^2 )Again, substitute s = sqrt(1 - x^2 )First component squared:(x -1 + s)^2 = (x -1)^2 + 2(x -1)s + s^2Second component squared:(s -1 -x)^2 = (s - (1 +x ))^2 = s^2 - 2s(1 +x ) + (1 +x )^2Therefore, |u|^2 = (x -1)^2 + 2(x -1)s + s^2 + s^2 - 2s(1 +x ) + (1 +x )^2Expand:= x^2 -2x +1 + 2(x -1)s + s^2 + s^2 - 2s(1 +x ) +1 + 2x +x^2Simplify:x^2 -2x +1 +1 +2x +x^2 = 2x^2 +22(x -1)s - 2s(1 +x ) = 2s(x -1 -1 -x ) = 2s(-2 ) = -4ss^2 + s^2 = 2s^2Therefore, |u|^2 = 2x^2 +2 -4s +2s^2But s^2 =1 -x^2, so:2x^2 +2 -4s +2(1 -x^2 ) =2x^2 +2 -4s +2 -2x^2 =4 -4sThus, |u| = sqrt(4 -4s )=2 sqrt(1 -s )Similarly, compute |v|:|v| = sqrt( (s -1)^2 + (-x )^2 ) = sqrt( (s -1)^2 +x^2 )Expand (s -1)^2:= s^2 -2s +1 +x^2But s^2 =1 -x^2, so:= (1 -x^2 ) -2s +1 +x^2 = 2 -2sTherefore, |v| = sqrt(2 -2s )=sqrt(2(1 -s ))=sqrt{2} sqrt(1 -s )Therefore, putting it all together:cosθ = (2(1 -s )) / (2 sqrt(1 -s ) * sqrt{2} sqrt(1 -s )) )Simplify denominator:2 * sqrt(1 -s ) * sqrt(2) * sqrt(1 -s )=2 * sqrt(2) * (1 -s )Numerator:2(1 -s )Therefore, cosθ = [2(1 -s )]/[2 sqrt(2)(1 -s ) ] =1/ sqrt(2 )Therefore, θ=45 degrees or π/4 radians.Wait, so the angle CDM is 45 degrees.But wait, this is regardless of the position of point C (as long as BC > AC, which in our coordinate system corresponds to x <0 ). Therefore, the angle CDM is always 45 degrees. So the answer is 45 degrees, which is π/4 radians. Therefore, the angle CDM is 45 degrees.But let me check with a specific example to ensure there's no mistake.Take point C at (-0.5, sqrt(1 -0.25)) = (-0.5, sqrt(0.75))≈(-0.5, 0.866)Compute point D:x-coordinate:1 - sqrt(1 - (-0.5)^2 )=1 - sqrt(0.75)≈1 -0.866≈0.134y-coordinate:1 + (-0.5)=0.5So D≈(0.134,0.5)Point M is (0,1)Compute vectors DC and DM.Vector DC: from D to C: (-0.5 -0.134, 0.866 -0.5 )≈(-0.634, 0.366 )Vector DM: from D to M: (0 -0.134,1 -0.5 )≈(-0.134,0.5 )Compute the angle between vectors (-0.634,0.366 ) and (-0.134,0.5 )Dot product: (-0.634)(-0.134) + (0.366)(0.5 )≈0.085 +0.183≈0.268Magnitude of DC: sqrt( (-0.634)^2 +0.366^2 )≈sqrt(0.401 +0.134 )≈sqrt(0.535 )≈0.731Magnitude of DM: sqrt( (-0.134)^2 +0.5^2 )≈sqrt(0.018 +0.25 )≈sqrt(0.268 )≈0.517Then cosθ≈0.268/(0.731*0.517 )≈0.268/0.378≈0.709Which is approximately 1/√2≈0.707, so close enough considering rounding errors. Therefore, θ≈45 degrees. So the calculation seems correct.Therefore, regardless of the position of point C (as long as BC > AC ), angle CDM is 45 degrees. Therefore, the answer is 45°, which is π/4 radians.Final Answerboxed{45^circ}</think>Given a right triangle (ABC) with (BC > AC), inscribed in a circle where (AB) is the hypotenuse and the diameter of the circle. A point (D) is chosen on (BC) such that (BD = AC), and point (M) is the midpoint of the arc (ACB). We need to find the angle (angle CDM).1. Circle and Midpoint (M): - Since (ABC) is a right triangle, the hypotenuse (AB) is the diameter of the circle, making the center of the circle the midpoint of (AB). - The midpoint of the arc (ACB) (which is a semicircle) is the point (M) diametrically opposite the right angle (C), which is at ((0, 1)) in a coordinate system where (A) and (B) are at ((-1, 0)) and ((1, 0)) respectively.2. Coordinates and Vectors: - Let (C) be ((x, y)) on the unit circle with (x < 0) (since (BC > AC)). - Point (D) is on (BC) such that (BD = AC). Using coordinate geometry, the coordinates of (D) are derived as ((1 - sqrt{1 - x^2}, 1 + x)).3. Vectors and Dot Product: - Vectors (DC) and (DM) are computed. - The dot product and magnitudes of these vectors are calculated to find the angle between them.4. Angle Calculation: - Using the dot product formula, the cosine of the angle between vectors (DC) and (DM) is found to be (frac{1}{sqrt{2}}), leading to the angle being (45^circ).Thus, the angle (angle CDM) is (boxed{45^circ}).]

🔑:Answer: 45^{circ}.Solution. Note that A B is the diameter of the circumscribed circle. Connect point M with points A, B, C, and D. Since arcs A M and B M are equal, the chords A M and B M that subtend them are also equal. Segments B D and A C are equal by the condition. Finally, angles M B C and M A C are equal, as inscribed angles subtending the same arc. Thus, triangles M B D and M A C are equal by two sides and the angle between them. This implies the equality of sides M D and M C, meaning that triangle M C D is isosceles. It remains to note that angle M C D=angle M C B=angle M A B=45^{circ}, since arc M B is half of the circle. Criteria for evaluation. Proof of the equality of triangles M B D and M A C - 3 points.

🧠:[<think>Okay, so I have this problem here about two sets, A and B. Set A has 100 elements, and set B has 50 elements. The question is about the number of mappings from A to B that satisfy two conditions: first, every element in B must have a preimage, meaning that for every b in B, there's at least one a in A such that f(a) = b. Second, the images of the elements of A under f must be non-decreasing. That is, f(a₁) ≤ f(a₂) ≤ ... ≤ f(a₁₀₀). The answer choices are given, and I need to figure out which one is correct.Let me start by understanding the problem step by step. So, we have a function f: A → B. Since A has 100 elements and B has 50, normally the number of functions would be 50^100, but here we have constraints. The first constraint is that every element in B must be hit by f; that is, f is surjective. The second constraint is that when we list the images of the elements of A in order, they form a non-decreasing sequence. Hmm. So, how do these two constraints interact? Let's consider the non-decreasing condition first. If the images are non-decreasing, then the function f is essentially determined by how many times each element of B appears in the sequence. Since the elements are ordered, once we know how many times each b_i is mapped to, the entire function is determined by the counts. For example, if we know that b₁ is mapped to by the first 5 elements, b₂ by the next 3, and so on, the non-decreasing order is maintained. But we also need to ensure that every b in B has at least one preimage. So, each b_i must be assigned at least one element from A. Therefore, this problem seems similar to distributing 100 distinct objects (the elements of A) into 50 distinct boxes (the elements of B) such that each box gets at least one object, and the order of the objects matters in a non-decreasing way. Wait, but in this case, the order is fixed once we assign how many elements go to each b_i. Because once we assign the number of elements that map to each b_i, the non-decreasing sequence is just grouping all the a_j's that map to b₁ first, then those that map to b₂, and so on. But since the elements of A are distinct, but the mapping is non-decreasing, the key is that the function is entirely determined by how we partition the 100 elements of A into 50 non-empty subsets, each corresponding to a preimage of an element in B. But wait, the elements of A are ordered, so the way we partition them must respect the order. That is, we have to split the ordered sequence a₁, a₂, ..., a₁₀₀ into 50 non-decreasing blocks, each assigned to a different b_i, and each block must be non-empty. However, since the function is non-decreasing, each block corresponds to a constant value in B, and the blocks must be ordered according to the order in B.But B is a set of real numbers, so they are ordered. The problem statement mentions "each element in B has a preimage," so we need to make sure that each element in B is assigned at least one element from A. So, the mapping is surjective, and non-decreasing. Therefore, the problem reduces to counting the number of surjective, non-decreasing functions from A to B. Since the function is non-decreasing, it's equivalent to finding the number of ways to partition the ordered set A into 50 ordered, non-empty intervals, each assigned to an element of B in order. But since B is a set of real numbers, the ordering of B must be considered. However, the problem doesn't specify the order of B; it's just a set. But since the function f must be non-decreasing, that implies that B is ordered. Wait, the problem statement says "the mapping f from A to B makes each element in B have a preimage, and f(a₁) ≤ f(a₂) ≤ ... ≤ f(a₁₀₀)". So, the order on B is induced by the function f. But B is a set of real numbers, which have a natural ordering. Therefore, the elements of B must be arranged in such a way that the order f(a₁) ≤ ... ≤ f(a₁₀₀) corresponds to the natural order of B. But since B is a set, it's just a collection of real numbers, but when we talk about the function f being non-decreasing, we have to map the elements of A to B in a way that the sequence f(a₁), ..., f(a₁₀₀) is non-decreasing. However, since B is a finite set, this sequence can only increase or stay the same as we go from a₁ to a₁₀₀.But to make sure that each element in B has a preimage, all elements of B must appear in the sequence. So, the problem is equivalent to counting the number of non-decreasing sequences of length 100 where each term is an element of B (which has 50 elements), and each element of B appears at least once. This is a classic stars and bars problem, but with the added twist that each element must appear at least once. In the standard stars and bars theorem, the number of non-decreasing sequences of length n with k possible values is C(n + k - 1, k - 1). But when we require that each value appears at least once, it's similar to the inclusion-exclusion principle. Wait, but perhaps there's a formula for that.Alternatively, think of it as a problem of placing dividers between the elements of A to assign them to the elements of B. Since the sequence is non-decreasing, once we assign a block of elements to a particular b_i, all subsequent blocks must be assigned to higher or equal b_j's. But since the function is surjective, each b_i must be assigned at least one element. So, the problem is equivalent to finding the number of compositions of 100 into 50 parts, each part at least 1. Because we need to divide 100 elements into 50 ordered non-empty subsets, which correspond to the number of ways to insert 49 dividers into the 99 gaps between the 100 elements. Wait, that would be C(99, 49). But wait, the standard formula for compositions is C(n - 1, k - 1). So, for dividing n elements into k parts, each at least 1, it's C(n - 1, k - 1). Here, n = 100, k = 50, so it would be C(99, 49). But the answer choices include C(99,49) as option D. But wait, let me verify.Wait, in the standard stars and bars problem, the number of ways to distribute n indistinct items into k distinct boxes, each with at least one item, is C(n - 1, k - 1). But in this case, the elements of A are ordered, and we need to assign them to the elements of B such that the assignment is non-decreasing. So, each such assignment corresponds to a composition of 100 into 50 parts, where each part is at least 1. The number of such compositions is indeed C(99, 49). But wait, the answer choices are given as:(A) C(100,50)(B) C(99,48)(C) C(100,49)(D) C(99,49)So, according to this, the answer should be D, C(99,49). But let me think again to be sure. Alternatively, another way to think about this is to imagine the 100 elements of A arranged in order. To create a non-decreasing function into B that is surjective, we need to partition the 100 elements into 50 non-empty intervals, each corresponding to an element of B. Since the function is non-decreasing, once we move to the next element in B, we can't go back. So, the problem is equivalent to inserting 49 dividers between the 100 elements to create 50 groups. Each divider can be placed in the gaps between elements. There are 99 gaps between the 100 elements. We need to choose 49 gaps to place the dividers, which is C(99,49). So, that would be option D. But let me check if the surjectivity is already considered here. Because if we just insert 49 dividers into the 99 gaps, that divides the 100 elements into 50 groups, each of which can be assigned to the 50 elements of B in order. Since each group is non-empty (because the dividers are placed between elements, so each group has at least one element), then each element of B is assigned at least one element of A. Therefore, this does account for the surjectivity, because each of the 50 groups must be assigned to a distinct element of B, and since each group is non-empty, each element of B has a preimage. Therefore, the number of such mappings is indeed C(99,49), which is option D. But wait, another thought: the elements of B are distinct real numbers, but when constructing the function f, the non-decreasing condition requires that once we switch to a higher b_i, we can't go back. However, the problem doesn't specify the order of B. But since f is non-decreasing, the images must be in non-decreasing order, so the order of B in the sequence must correspond to their natural order as real numbers. Since B is a set of real numbers, their natural order is fixed. Therefore, the mapping f must assign the first group to the smallest element in B, the next group to the next smallest, etc. But since the problem doesn't specify which element of B is which, just that B is a set, does that affect the count?Wait, the problem says that the mapping f is from A to B, with each element in B having a preimage, and the images are non-decreasing. Since B is a set of real numbers, their order is inherent. So, the function f must map elements of A to B such that the sequence f(a₁), ..., f(a₁₀₀) is non-decreasing and covers all elements of B. However, the actual labels of the elements of B (i.e., which one is b₁, b₂, etc.) could affect the count. But in the problem statement, B is just presented as a set, so unless specified otherwise, we can assume that the order of B is arbitrary. However, in the answer options, the numbers given don't involve permutations of B. Wait, but in our previous reasoning, we assumed that once you partition A into 50 groups, each group is assigned to a distinct element of B in order. But if the order of B is not fixed, then perhaps we would have to multiply by the number of permutations of B. However, the problem states that the mapping f must make each element in B have a preimage. So, if we consider that the elements of B are distinguishable, then the assignment of the groups to the elements of B would matter. But in our previous calculation, we assumed that the order is determined by the non-decreasing sequence, so that the first group maps to the smallest element of B, the second group to the next smallest, etc. However, since the problem doesn't specify the order of B, but B is a set of real numbers, which have a natural order, the function f must map to them in a way that is consistent with their natural order. Therefore, the labels b₁, b₂, ..., b₅₀ must be in increasing order. Wait, the problem statement says "B = {b₁, b₂, ..., b₅₀}", but since it's a set, the indices are just for labeling. However, in the context of real numbers, the subscripts might be in order. Wait, the problem doesn't specify whether the elements of B are ordered in any particular way. This is a bit confusing. If the elements of B are not ordered, but the function f must be non-decreasing, which requires that the images are ordered, then we have to assign the groups to the elements of B such that the order is respected. However, since B is a set of real numbers, they have a natural order. So, regardless of how we label the elements of B, there is an inherent order. Therefore, the mapping f is required to arrange the images in a non-decreasing sequence, which must correspond to the natural order of B. Therefore, the labels of B must be such that b₁ < b₂ < ... < b₅₀, otherwise, the non-decreasing function f couldn't map to all elements of B. Wait, actually, no. If B is just a set of real numbers without a specified order, then the function f can map to them in any order, but the sequence f(a₁), ..., f(a₁₀₀) must be non-decreasing. However, since all elements of B must be used, the images must consist of all elements of B arranged in some order. But in a non-decreasing sequence, the elements must be arranged from the smallest to the largest. Therefore, the function f must map the first group of A to the smallest element of B, the next group to the next smallest, etc. Therefore, the mapping is uniquely determined by the partition of A into 50 groups and the order of B. But since B is a set of real numbers, their order is fixed. Therefore, the labels of B (b₁, b₂, ..., b₅₀) might not necessarily correspond to their order. Wait, but if the problem doesn't specify that the elements of B are listed in order, then we cannot assume that b₁ < b₂ < ... < b₅₀. However, since the problem is presented with B as a set, the order of the labels is irrelevant. This is a critical point. If B is just a set, then the labels b₁, b₂, ..., b₅₀ don't imply any order. However, the function f has to be non-decreasing in the sense that f(a₁) ≤ f(a₂) ≤ ... ≤ f(a₁₀₀). Therefore, the images must be a non-decreasing sequence of elements from B, which as real numbers, have their own natural order. Therefore, to have a non-decreasing sequence covering all elements of B, the images must include each element of B at least once, and the sequence must be ordered according to the natural order of B. However, the problem doesn't specify the order of B, so we need to account for the fact that the labels of B could be permuted. Wait, but the answer choices don't include a factorial term, which would be necessary if we had to consider permutations of B. Therefore, perhaps in this problem, the elements of B are considered to be ordered as b₁ ≤ b₂ ≤ ... ≤ b₅₀. Otherwise, the problem is ill-posed. Given that the answer choices are combinatorial numbers without factorials, it's likely that the order of B is fixed. For example, if B is considered as an ordered set, with b₁ < b₂ < ... < b₅₀, then the function f must map elements of A to B in a non-decreasing sequence, hitting each b_i at least once. In this case, the number of such functions would be the same as the number of ways to partition the 100 elements of A into 50 non-empty intervals, each assigned to a specific b_i in order. Therefore, the count is C(99,49), as previously reasoned. But let me confirm this with another approach. Suppose we model each such function as a sequence of 100 elements in B that is non-decreasing and surjective. The number of such sequences is equal to the number of ways to distribute 100 indistinguishable items into 50 distinguishable boxes, each box containing at least one item, which is C(100 - 1, 50 - 1) = C(99,49). However, in our problem, the items (the elements of A) are distinguishable, but because the function is non-decreasing, the specific order doesn't matter beyond the requirement that the sequence is non-decreasing. Wait, but in the case of distinguishable elements, the number of functions would normally be higher. But in our case, since the function has to be non-decreasing, the assignment is determined entirely by the counts of how many elements map to each b_i. Wait, no, because the elements of A are distinguishable, the actual assignment depends on which elements go to which b_i. However, the non-decreasing condition imposes that once you choose the multiset of b_i's (with each b_i appearing at least once), the order is fixed. Therefore, the problem reduces to choosing how many elements map to each b_i, with each count at least 1, and the sum of counts being 100. But since the elements of A are distinguishable, the number of such functions would be the same as the number of ways to assign each element of A to a b_i such that each b_i gets at least one element, and the assignment is non-decreasing. But because the elements are ordered, the assignment is determined by choosing positions where the function increases. Wait, perhaps another way: the number of non-decreasing surjective functions from A to B is equal to the number of ordered partitions of A into 50 non-empty subsets, where the order of the subsets corresponds to the order of B. Since A is ordered, the number of ways to partition A into 50 non-empty intervals is C(99,49). Each interval corresponds to a group of elements mapped to a particular b_i in order. Therefore, this seems correct. Alternatively, think of it as arranging 100 elements in a row and placing 49 dividers between them to create 50 groups. Each group must have at least one element, so the number of ways is C(99,49). Each group is then assigned to b₁, b₂, ..., b₅₀ in order, ensuring that each b_i has at least one preimage, and the function is non-decreasing. Since the elements of A are in a fixed order, once we decide where the dividers go, the function is determined. Therefore, the number of such mappings is indeed C(99,49), which is option D. But let me check with a smaller example. Suppose A has 3 elements and B has 2 elements. Then the number of surjective non-decreasing functions should be C(2,1) = 2. Let's verify. The possible functions would be:1. Assign a₁ and a₂ to b₁, and a₃ to b₂. The sequence is b₁, b₁, b₂.2. Assign a₁ to b₁, and a₂ and a₃ to b₂. The sequence is b₁, b₂, b₂.So there are 2 such functions, which is C(3-1, 2-1) = C(2,1) = 2. So that works. Similarly, if A has n elements and B has k elements, the number of surjective non-decreasing functions is C(n-1, k-1). Therefore, in this case, n=100, k=50, so C(99,49), which is option D. Therefore, after considering various approaches and verifying with a smaller example, I believe the correct answer is D, C(99,49).Final Answerboxed{D}</think>Given two sets of real numbers ( A = {a_1, a_2, ldots, a_{100}} ) and ( B = {b_1, b_2, ldots, b_{50}} ), we need to find the number of surjective mappings ( f ) from ( A ) to ( B ) such that ( f(a_1) leqslant f(a_2) leqslant cdots leqslant f(a_{100}) ).1. Surjective and Non-decreasing Function: The function ( f ) must be surjective, meaning each element in ( B ) must have at least one preimage in ( A ). Additionally, the sequence ( f(a_1), f(a_2), ldots, f(a_{100}) ) must be non-decreasing.2. Ordered Partition: The non-decreasing condition implies that the elements of ( A ) can be partitioned into 50 non-empty intervals, each corresponding to an element of ( B ). These intervals must be ordered such that the images correspond to the natural order of ( B ).3. Combinatorial Analysis: To partition 100 elements into 50 non-empty intervals, we need to place 49 dividers between the elements. The number of ways to place these dividers in the 99 gaps between 100 elements is given by the binomial coefficient ( binom{99}{49} ).4. Verification with Smaller Example: Considering a smaller example with ( A ) having 3 elements and ( B ) having 2 elements, the number of surjective non-decreasing functions is verified to be ( binom{2}{1} = 2 ), which matches the formula ( binom{n-1}{k-1} ).Thus, the number of such mappings is ( binom{99}{49} ), which corresponds to option D.[boxed{D}],<think>Okay, let's try to figure out this problem. So we have two sets, A with 100 elements and B with 50 elements. We need to find the number of mappings f from A to B such that each element in B has a preimage, meaning that for every b in B, there's at least one a in A with f(a) = b. Additionally, the function f has to be non-decreasing, right? Because it says f(a₁) ≤ f(a₂) ≤ ... ≤ f(a₁₀₀). Hmm, okay. So first, since the function is non-decreasing, the order of the elements in A matters. Each element in A has to map to an element in B such that the sequence of images is non-decreasing. Also, every element in B must be mapped to by at least one element in A. So, if I think about this, it's similar to arranging the 100 elements of A into 50 ordered bins (since B has 50 elements), where each bin must have at least one element. Because the function is non-decreasing, the order of the elements in A must be preserved when assigning them to the bins. So the first bin will correspond to the smallest element in B, the next bin to the next smallest, etc. Wait, but in this case, the elements of B are just labeled as b₁, b₂, ..., b₅₀. The problem doesn't specify any order on B, but the function f is mapping to B such that the images are non-decreasing. So I suppose we can assume that the ordering of B is given by their indices, i.e., b₁ ≤ b₂ ≤ ... ≤ b₅₀. Otherwise, the problem wouldn't make sense because the function's images need to be non-decreasing. So the mapping has to take the elements of A and assign them to elements in B in a way that the sequence is non-decreasing, and each b in B is hit at least once.This seems like a classic stars and bars problem, but with the added constraint that each bin (each element of B) must have at least one element. In the stars and bars theorem, the number of ways to distribute n indistinct objects into k distinct bins is C(n + k - 1, k - 1). If each bin must have at least one object, it becomes C(n - 1, k - 1). But wait, in this case, the elements of A are ordered, and we need to preserve their order when assigning them to B. Wait a second, actually, the non-decreasing condition is equivalent to arranging the elements of A into B such that if you go from a₁ to a₁₀₀, each subsequent element is mapped to the same or a higher element in B. So this is similar to dividing the 100 elements into 50 ordered groups, where each group corresponds to an element of B, and each group must have at least one element. The number of such groupings would then be the same as the number of compositions of 100 into 50 parts, each at least 1. But compositions are ordered, right? So the number of compositions of n into k parts is C(n - 1, k - 1). So in this case, n = 100, k = 50, so the number would be C(99, 49). Hmm, but let me verify this. If we have 100 elements in a row, and we need to split them into 50 groups, each with at least one element, we need to place 49 dividers between the elements. Since there are 99 gaps between the 100 elements, the number of ways to place 49 dividers is C(99, 49). That's the standard stars and bars approach when each bin must have at least one object. So in this case, since we need each element of B to have at least one preimage, we need to partition the 100 elements into 50 non-empty subsets, ordered by the order of B. Therefore, the number of such mappings should be C(99, 49). Looking at the options, that's answer choice D: C_{99}^{49}. Wait, but let me double-check. Another way to think about it: if we have a non-decreasing function from A to B where each element of B is hit at least once, then the problem is equivalent to finding the number of surjective non-decreasing functions. In combinatorics, the number of non-decreasing functions from a totally ordered set of size n to a totally ordered set of size k is C(n + k - 1, k - 1). However, if the function is required to be surjective, then it's equivalent to C(n - 1, k - 1). Wait, let me confirm. Suppose we have n elements in the domain and k elements in the codomain. A non-decreasing function is equivalent to a multiset of size n where each element is from the codomain. The number of such multisets is C(n + k - 1, k - 1). But if we require that each element in the codomain appears at least once, then it's equivalent to a composition of n into k parts, each at least 1, which is C(n - 1, k - 1). Yes, so in this case, n = 100, k = 50. So the number of surjective non-decreasing functions is C(100 - 1, 50 - 1) = C(99, 49), which is option D. But just to ensure that I haven't mixed up the formula, let me recall. The formula for the number of non-decreasing surjective functions from an ordered domain of size n to an ordered codomain of size k is the same as the number of ordered partitions of the domain into k non-empty intervals, which is indeed C(n - 1, k - 1). Each such partition corresponds to inserting k - 1 dividers between the n elements, which can be done in C(n - 1, k - 1) ways. Therefore, the correct answer is C(99, 49), which is option D. So, I think the answer is D.Final Answerboxed{D}</think>Given two sets of real numbers ( A = {a_1, a_2, ldots, a_{100}} ) and ( B = {b_1, b_2, ldots, b_{50}} ), we need to find the number of mappings ( f ) from ( A ) to ( B ) such that each element in ( B ) has a preimage and the sequence ( f(a_1) leqslant f(a_2) leqslant cdots leqslant f(a_{100}) ) is non-decreasing.To solve this, we recognize that the non-decreasing condition implies the elements of ( A ) must be partitioned into 50 non-empty intervals, each corresponding to an element of ( B ). This is equivalent to finding the number of compositions of 100 elements into 50 parts, each part being at least 1. Using the stars and bars theorem, the number of ways to distribute 100 elements into 50 non-empty intervals is given by the binomial coefficient ( binom{99}{49} ). This is because we need to place 49 dividers between the 100 elements, which can be done in ( binom{99}{49} ) ways.Thus, the number of such mappings is ( boxed{D} ).]

🔑:Let's assume b_{1}<b_{2}<cdots<b_{50}.Since each element in set B has a preimage, let the set of preimages of b_{i} be A_{i}(i=1,2, cdots, 50), with the number of elements being x_{i}. Thenx_{1}+x_{2}+cdots+x_{50}=100 text {. }Thus, the problem is transformed into finding the number of positive integer solutions to the indeterminate equation (1), which has mathrm{C}_{99}^{49} solutions. Therefore, the correct choice is (D).

🧠:[<think>Okay, so I need to solve this problem where I have to find the equation of a line l that passes through the intersection point P of two given lines and is perpendicular to another given line. Then, I also have to find the area of the triangle formed by line l and the coordinate axes. Let me try to break this down step by step.First, part (1) asks for the equation of line l. The line l passes through the intersection point P of the lines 3x + 4y - 2 = 0 and 2x + y + 2 = 0. Also, line l is perpendicular to the line x - 2y - 1 = 0. So, my first task is to find the coordinates of point P, which is the intersection of the two lines given. Once I have point P, I need to determine the slope of line l, knowing that it's perpendicular to x - 2y - 1 = 0. Then, using the point-slope form, I can write the equation of line l.Let me start by finding the intersection point P of 3x + 4y - 2 = 0 and 2x + y + 2 = 0.To find the intersection of two lines, I can solve the system of equations:1) 3x + 4y = 22) 2x + y = -2Maybe I can use substitution or elimination. Let's try elimination. If I multiply the second equation by 4, I can eliminate y.Original equations:3x + 4y = 2 ...(1)2x + y = -2 ...(2)Multiply equation (2) by 4: 8x + 4y = -8 ...(2a)Now subtract equation (1) from (2a):(8x + 4y) - (3x + 4y) = -8 - 28x - 3x + 4y - 4y = -105x = -10So x = -10 / 5 = -2Now substitute x = -2 into equation (2) to find y:2*(-2) + y = -2-4 + y = -2y = -2 + 4 = 2So the intersection point P is (-2, 2). Let me check if this point satisfies both original equations.Check equation 1: 3*(-2) + 4*2 = -6 + 8 = 2, which matches the right-hand side of equation 1.Check equation 2: 2*(-2) + 2 = -4 + 2 = -2, which matches the right-hand side of equation 2. So P is indeed (-2, 2).Now, line l is perpendicular to the line x - 2y - 1 = 0. Let me find the slope of the line x - 2y - 1 = 0 first.Rewriting x - 2y - 1 = 0 into slope-intercept form:x - 1 = 2yy = (1/2)x - 1/2So the slope (m1) of this line is 1/2. The line l is perpendicular to this, so its slope (m2) should be the negative reciprocal of 1/2, which is -2.Therefore, the slope of line l is -2.Now that I have the slope of line l and a point it passes through (P(-2, 2)), I can use the point-slope form to write the equation.Point-slope form: y - y1 = m(x - x1)Substituting m = -2, x1 = -2, y1 = 2:y - 2 = -2(x - (-2)) => y - 2 = -2(x + 2)Expanding the right-hand side:y - 2 = -2x - 4Bring all terms to the left-hand side:y - 2 + 2x + 4 = 0 => 2x + y + 2 = 0Wait, hold on. Let me check that again. If I expand y - 2 = -2x - 4, then adding 2 to both sides gives y = -2x - 2. So the equation in slope-intercept form is y = -2x - 2. Converting this to standard form:2x + y + 2 = 0.Wait, but let's check if this line passes through point P(-2, 2). Plugging x = -2, y = 2 into 2x + y + 2: 2*(-2) + 2 + 2 = -4 + 2 + 2 = 0. So yes, it does pass through P. And the slope is -2, which is correct because it's perpendicular to the line with slope 1/2. So part (1) seems to be done. The equation of line l is 2x + y + 2 = 0.But wait, let me confirm once more. The original lines were 3x + 4y - 2 = 0 and 2x + y + 2 = 0. Their intersection is at (-2, 2). Then the line l is perpendicular to x - 2y -1 = 0, which has slope 1/2, so line l has slope -2. The equation we found is 2x + y + 2 = 0. Let me check the slope: rewriting as y = -2x - 2, so slope is indeed -2. And passes through (-2, 2): y = -2*(-2) - 2 = 4 - 2 = 2. Correct. So yes, part (1) answer is 2x + y + 2 = 0. Alternatively, it can be written as 2x + y = -2, but the standard form is usually written as ax + by + c = 0, so 2x + y + 2 = 0.Moving on to part (2): Calculate the area S of the triangle formed by line l and the two coordinate axes.To find the area of the triangle formed by line l with the x-axis and y-axis, I need to find the x-intercept and y-intercept of line l, then use the formula for the area of a right-angled triangle: (1/2)*base*height, where base and height are the intercepts.First, let's find the x-intercept of line l. The x-intercept is where y = 0. Substitute y = 0 into the equation of line l:2x + 0 + 2 = 0 => 2x + 2 = 0 => 2x = -2 => x = -1. So the x-intercept is (-1, 0).Next, find the y-intercept. The y-intercept is where x = 0. Substitute x = 0 into the equation:2*0 + y + 2 = 0 => y + 2 = 0 => y = -2. So the y-intercept is (0, -2).Therefore, the triangle is formed by the points (-1, 0), (0, -2), and the origin (0, 0), since the axes intercepts are at (-1, 0) and (0, -2). Wait, but actually, the triangle is formed by the intercepts and the origin. So the vertices are (0,0), (-1,0), and (0,-2). Let me visualize this. The line crosses the x-axis at (-1, 0) and the y-axis at (0, -2). Connecting these two intercepts with the origin forms a right triangle with legs along the axes. The base can be considered as the distance from (-1, 0) to (0, 0), which is 1 unit, and the height as the distance from (0, -2) to (0, 0), which is 2 units. However, since intercepts can be positive or negative, but lengths are positive. So the base length is 1, the height is 2. So the area would be (1/2)*1*2 = 1.Wait, but let me confirm. The x-intercept is at x = -1, so the length from the origin to the x-intercept is | -1 | = 1. Similarly, the y-intercept is at y = -2, so the length from the origin to the y-intercept is | -2 | = 2. So indeed, the area is (1/2)*1*2 = 1.But wait, hold on. The line crosses the x-axis at (-1, 0) and y-axis at (0, -2). The triangle formed by these two intercepts and the axes would have vertices at (-1, 0), (0, -2), and (0, 0). The base is along the x-axis from (0,0) to (-1,0), length 1. The height is along the y-axis from (0,0) to (0,-2), length 2. So area is (1/2)*base*height = (1/2)*1*2 = 1. So the area S is 1. Therefore, the answer for part (2) is 1.But let me double-check. Alternatively, using the intercept form of the line. The intercept form is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. So line l is 2x + y + 2 = 0. Let's write it as 2x + y = -2. Divide both sides by -2 to get x/(-1) + y/(-2) = 1. So the x-intercept is -1 and y-intercept is -2. Therefore, the intercepts are at (-1, 0) and (0, -2), same as before. So the area is indeed (1/2)*|a|*|b| = (1/2)*1*2 = 1. Yep, that checks out.Wait, but let me also compute the area using coordinates. The three vertices of the triangle are (0,0), (-1,0), and (0,-2). Using the formula for the area of a triangle with coordinates (x1,y1), (x2,y2), (x3,y3):Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Plugging in (0,0), (-1,0), (0,-2):Area = |0*(0 - (-2)) + (-1)*(-2 - 0) + 0*(0 - 0)| / 2= |0 + (-1)*(-2) + 0| / 2= |0 + 2 + 0| / 2= 2 / 2 = 1. So same result. Therefore, the area is indeed 1.So summarizing:(1) The equation of line l is 2x + y + 2 = 0.(2) The area of the triangle is 1.But wait, the problem says "the area S of the triangle formed by line l and the two coordinate axes." Just to confirm once again. The line cuts the axes at (-1,0) and (0,-2). The triangle is formed with the axes, so those two intercept points and the origin. So yes, the triangle is a right triangle with legs of lengths 1 and 2, so area 1. That seems correct.Therefore, I think these are the correct answers.Final Answer(1) The equation of line l is boxed{2x + y + 2 = 0}.(2) The area S of the triangle is boxed{1}.</think>To find the equation of line ( l ) that passes through the intersection point ( P ) of the lines ( 3x + 4y - 2 = 0 ) and ( 2x + y + 2 = 0 ), and is perpendicular to the line ( x - 2y - 1 = 0 ):1. Find the intersection point ( P ): - Solve the system of equations: [ begin{cases} 3x + 4y = 2 2x + y = -2 end{cases} ] - Multiply the second equation by 4: ( 8x + 4y = -8 ) - Subtract the first equation from the result: ( 5x = -10 Rightarrow x = -2 ) - Substitute ( x = -2 ) into ( 2x + y = -2 ): ( -4 + y = -2 Rightarrow y = 2 ) - Intersection point ( P ) is ( (-2, 2) ).2. Determine the slope of line ( l ): - The line ( x - 2y - 1 = 0 ) has a slope of ( frac{1}{2} ). - The slope of line ( l ), being perpendicular, is the negative reciprocal: ( -2 ).3. Equation of line ( l ): - Using point-slope form with point ( P(-2, 2) ) and slope ( -2 ): [ y - 2 = -2(x + 2) implies y - 2 = -2x - 4 implies 2x + y + 2 = 0 ]To calculate the area ( S ) of the triangle formed by line ( l ) and the coordinate axes:1. Find intercepts of line ( l ): - x-intercept: Set ( y = 0 ): [ 2x + 0 + 2 = 0 implies 2x = -2 implies x = -1 ] x-intercept is ( (-1, 0) ). - y-intercept: Set ( x = 0 ): [ 2(0) + y + 2 = 0 implies y = -2 ] y-intercept is ( (0, -2) ).2. Calculate the area: - The triangle vertices are ( (0, 0) ), ( (-1, 0) ), and ( (0, -2) ). - Using the formula for the area of a right triangle: [ text{Area} = frac{1}{2} times text{base} times text{height} = frac{1}{2} times 1 times 2 = 1 ] Final Answer(1) The equation of line ( l ) is boxed{2x + y + 2 = 0}.(2) The area ( S ) of the triangle is boxed{1}.,<think>Okay, let's tackle this problem step by step. First, I need to find the equation of line ( l ) which passes through the intersection point ( P ) of the two given lines ( 3x + 4y - 2 = 0 ) and ( 2x + y + 2 = 0 ), and is also perpendicular to the line ( x - 2y - 1 = 0 ). Then, I have to calculate the area of the triangle formed by line ( l ) and the coordinate axes. Let me start with part (1).Problem (1): Find the equation of line ( l ).First, I need to find the point ( P ), the intersection of the two lines ( 3x + 4y - 2 = 0 ) and ( 2x + y + 2 = 0 ). To do this, I'll solve these two equations simultaneously.Let me write down the equations:1. ( 3x + 4y = 2 ) (Equation A)2. ( 2x + y = -2 ) (Equation B)I can use the substitution or elimination method. Let's try elimination. If I multiply Equation B by 4 to make the coefficients of ( y ) the same as in Equation A:Multiply Equation B by 4:( 8x + 4y = -8 ) (Equation C)Now, subtract Equation A from Equation C:( (8x + 4y) - (3x + 4y) = -8 - 2 )Simplify:( 5x = -10 )So, ( x = -10/5 = -2 ).Now substitute ( x = -2 ) back into Equation B to find ( y ):Equation B: ( 2(-2) + y = -2 )Calculate:( -4 + y = -2 )Add 4 to both sides:( y = 2 ).So, the intersection point ( P ) is ( (-2, 2) ). Let me verify that this point satisfies both original equations.Check Equation A: ( 3(-2) + 4(2) = -6 + 8 = 2 ). Correct, since Equation A is ( 3x +4y =2 ).Check Equation B: ( 2(-2) + 2 = -4 + 2 = -2 ). Correct, since Equation B is ( 2x + y = -2 ).Good, so ( P(-2, 2) ) is correct.Next, line ( l ) is perpendicular to the line ( x - 2y -1 = 0 ). Let's find the slope of the given line ( x - 2y -1 = 0 ).First, rewrite it in slope-intercept form ( y = mx + b ):( x - 2y -1 = 0 )Add ( 2y ) to both sides:( x -1 = 2y )Divide both sides by 2:( y = (1/2)x - 1/2 )So, the slope ( m ) of this line is ( 1/2 ).Since line ( l ) is perpendicular to this line, its slope ( m_l ) must be the negative reciprocal of ( 1/2 ), which is ( -2 ).Therefore, the slope of line ( l ) is ( -2 ).Now, we know that line ( l ) passes through point ( P(-2, 2) ) and has slope ( -2 ). Let's use the point-slope form to write its equation.Point-slope form is:( y - y_1 = m(x - x_1) )Plugging in ( m = -2 ), ( x_1 = -2 ), ( y_1 = 2 ):( y - 2 = -2(x - (-2)) )Simplify the equation:( y - 2 = -2(x + 2) )Distribute the -2 on the right:( y - 2 = -2x - 4 )Add 2 to both sides:( y = -2x - 4 + 2 )Simplify:( y = -2x - 2 )Alternatively, converting this to standard form ( Ax + By + C = 0 ):Bring all terms to the left:( 2x + y + 2 = 0 )Wait a second, but the original lines given were ( 3x +4y -2 =0 ) and ( 2x + y +2=0 ). The line we found here, ( 2x + y +2=0 ), is actually one of the original lines. But that can't be right, because line ( l ) is supposed to pass through their intersection point and be perpendicular to ( x -2y -1 =0 ). But line ( 2x + y +2 =0 ) is one of the original lines. However, since line ( l ) is perpendicular to ( x -2y -1=0 ), which has slope 1/2, so line l has slope -2. But the line 2x + y +2 =0 also has slope -2. Wait, that's correct, because 2x + y +2 =0 can be written as y = -2x -2, which has slope -2. But that line is one of the original two lines given. However, their intersection point is (-2,2), which lies on both lines. So line l is coinciding with one of the original lines?Wait, but line l is supposed to pass through point P and be perpendicular to x -2y -1=0. The line 2x + y +2=0 does pass through P and is perpendicular to x -2y -1=0. Let me check if the slope is indeed -2.Yes, as we found, the slope of line l is -2, which matches the slope of 2x + y +2=0. So actually, line l is the same as the line 2x + y +2=0. But wait, the problem states that line l passes through the intersection point P of the two lines. Since line 2x + y +2=0 is one of those lines, its intersection with the other line is point P. So line l is coinciding with one of the original lines. Is that possible?Wait, maybe. Let me verify if 2x + y +2=0 is indeed perpendicular to x -2y -1=0.Slope of x -2y -1=0 is 1/2. Slope of 2x + y +2=0 is -2. The product of the slopes is (1/2)*(-2) = -1, which confirms that they are perpendicular. So yes, line l is indeed the line 2x + y +2=0, which passes through point P (as it should, since P is the intersection of 3x +4y -2=0 and 2x + y +2=0). Therefore, the equation of line l is 2x + y +2=0. But let me check again if there's a miscalculation.Wait, when we derived the equation of line l, we started with point P(-2, 2) and slope -2, leading us to y = -2x -2, which is equivalent to 2x + y +2=0. That's correct. So even though it coincides with one of the original lines, that's okay because that line does pass through the intersection point and has the required slope.Therefore, the answer to part (1) is ( 2x + y + 2 = 0 ), or in standard form. Alternatively, maybe the problem expects a simplified version, but since 2x + y +2=0 is already in standard form, that's acceptable.Problem (2): Calculate the area ( S ) of the triangle formed by line ( l ) and the two coordinate axes.To find the area of the triangle formed by line ( l ) and the coordinate axes, we need to find the x-intercept and y-intercept of line ( l ), then use the formula for the area of a right-angled triangle: ( S = frac{1}{2} times |xtext{-intercept}| times |ytext{-intercept}| ).First, let's find the intercepts of line ( l ): equation ( 2x + y + 2 = 0 ).Finding the x-intercept: Set ( y = 0 ):( 2x + 0 + 2 = 0 )( 2x + 2 = 0 )( 2x = -2 )( x = -1 )So, the x-intercept is at (-1, 0).Finding the y-intercept: Set ( x = 0 ):( 2(0) + y + 2 = 0 )( y + 2 = 0 )( y = -2 )So, the y-intercept is at (0, -2).Now, the triangle is formed by the points (-1, 0), (0, -2), and the origin (0, 0), but wait, actually, the triangle is formed by the intercepts and the origin. Wait, no. The line intersects the x-axis at (-1, 0) and the y-axis at (0, -2). The triangle formed by the line and the axes is the triangle with vertices at (-1, 0), (0, -2), and the origin (0,0). Wait, but actually, the axes themselves are the x-axis and y-axis, so the intercepts are (-1,0) and (0,-2), and the triangle is between these two intercepts and the origin. So yes, the three vertices are (0,0), (-1,0), and (0,-2). But since intercepts can be positive or negative, but the length is absolute.But the area is calculated as ( frac{1}{2} times |x-intercept| times |y-intercept| ).So, here, x-intercept is -1, so | -1 | = 1. y-intercept is -2, so | -2 | = 2. Therefore, area ( S = frac{1}{2} times 1 times 2 = 1 ).Wait, let me confirm. The triangle's base can be along the x-axis from (0,0) to (-1,0), which is length 1. The height is along the y-axis from (0,0) to (0,-2), length 2. Since it's a right-angled triangle, the area is indeed (1 * 2)/2 = 1. So area ( S = 1 ).But let me verify this by plotting or using coordinates.The three points are (0,0), (-1,0), and (0,-2). Using the formula for the area of a triangle with coordinates:Area = ( frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| )Plugging in (0,0), (-1,0), (0,-2):( frac{1}{2} |0*(0 - (-2)) + (-1)*(-2 - 0) + 0*(0 - 0)| )= ( frac{1}{2} |0 + (-1)*(-2) + 0| )= ( frac{1}{2} |0 + 2 + 0| = frac{1}{2} times 2 = 1 ). Correct.So the area is indeed 1. Therefore, the answer for part (2) is 1.Wait, but hold on. Let me check again if the intercepts are correct. For line ( l ): 2x + y + 2 = 0.x-intercept when y=0: 2x + 0 + 2 = 0 → 2x = -2 → x = -1. Correct.y-intercept when x=0: 0 + y + 2 = 0 → y = -2. Correct.So intercepts at (-1, 0) and (0, -2). The triangle is between these two points and the origin. The area is 1. Seems correct.But just to be thorough, let me visualize the line. It's going from (-1, 0) to (0, -2). The triangle is in the fourth quadrant, but area is positive. So yes, 1 is correct.Final Answers(1) The equation of line ( l ) is ( 2x + y + 2 = 0 ).(2) The area ( S ) of the triangle is 1.But wait, let me cross-verify part (1). When we derived the equation of line ( l ), we used point-slope form and arrived at ( y = -2x -2 ), which is indeed equivalent to ( 2x + y + 2 = 0 ). But the original line given in the problem was ( 2x + y + 2 = 0 ), which is one of the lines whose intersection we found. So line ( l ) is actually the same as that original line. But that seems odd, but mathematically, since that line does pass through point P (as P is the intersection of the two lines, including this one), and it is indeed perpendicular to the line ( x - 2y -1 =0 ), because their slopes multiply to -1. So even though it's one of the original lines, it satisfies all the conditions. So the answer is correct.Another way to think about it: If two lines intersect at point P, then any line passing through P can be a different line, unless it's one of the original lines. In this case, the line ( 2x + y +2 =0 ) passes through P and is perpendicular to the given third line. So even if it's one of the original lines, it's still a valid answer. So part (1) is correct.Therefore, the final answers are:(1) ( 2x + y + 2 = 0 )(2) ( S = 1 )Final Answer(1) boxed{2x + y + 2 = 0}(2) boxed{1}</think>To find the equation of line ( l ) that passes through the intersection point ( P ) of the lines ( 3x + 4y - 2 = 0 ) and ( 2x + y + 2 = 0 ) and is perpendicular to the line ( x - 2y - 1 = 0 ):1. Finding the intersection point ( P ): - Solve the system of equations: [ begin{cases} 3x + 4y = 2 2x + y = -2 end{cases} ] - Multiply the second equation by 4: ( 8x + 4y = -8 ) - Subtract the first equation from this result: ( 5x = -10 implies x = -2 ) - Substitute ( x = -2 ) back into the second equation: ( 2(-2) + y = -2 implies y = 2 ) - Intersection point ( P ) is ( (-2, 2) ).2. Slope of the line perpendicular to ( x - 2y - 1 = 0 ): - Rewrite ( x - 2y - 1 = 0 ) in slope-intercept form: ( y = frac{1}{2}x - frac{1}{2} ) - The slope is ( frac{1}{2} ), so the perpendicular slope is ( -2 ).3. Equation of line ( l ): - Using point-slope form with point ( P(-2, 2) ) and slope ( -2 ): [ y - 2 = -2(x + 2) implies y = -2x - 2 ] - Convert to standard form: ( 2x + y + 2 = 0 ).To calculate the area ( S ) of the triangle formed by line ( l ) and the coordinate axes:1. Finding intercepts: - x-intercept: Set ( y = 0 ): ( 2x + 2 = 0 implies x = -1 ) - y-intercept: Set ( x = 0 ): ( y + 2 = 0 implies y = -2 )2. Area calculation: - The area is ( frac{1}{2} times |xtext{-intercept}| times |ytext{-intercept}| ): [ S = frac{1}{2} times 1 times 2 = 1 ] Final Answers(1) boxed{2x + y + 2 = 0}(2) boxed{1}]

🔑:(1) From the system of equationsbegin{cases}3x+4y-2=0 2x+y+2=0end{cases},we solve for x and y and obtainbegin{cases}x=-2 y=2end{cases}.Therefore, the coordinates of point P are (-2,2).Since point P has the coordinates (-2,2), and line l is perpendicular to the line x-2y+1=0, we can assume that the equation of line l has the form 2x+y+C=0.Substituting the coordinates of point P into the equation gives us:2cdot(-2) + 2 + C = 0,which simplifies to C = 2.Therefore, the equation of line l is boxed{2x + y + 2 = 0}.(2) From the equation of line l, we know that its intercepts on the x-axis and y-axis are -1 and -2, respectively.Thus, the area S of the triangle formed by line l and the two coordinate axes can be calculated asS=frac{1}{2}times 1times 2=boxed{1}.