🧠:[<think>Okay, let's tackle this trapezoid problem. So, we have trapezoid ABCD with AB parallel to DC. M is the midpoint of CD. There are some perpendicularity conditions: AD is perpendicular to CD, AC is perpendicular to BM, and BC is perpendicular to BD. We need to find the ratio AB/CD.First, I should probably sketch the trapezoid to visualize the problem. Let me imagine it: AB is the top base, DC is the bottom base, and since it's a trapezoid with AB parallel to DC, the legs are AD and BC. But wait, AD is perpendicular to CD, so that means AD is a vertical leg if CD is the base. So, maybe this is a right trapezoid at D? Because AD is perpendicular to DC. But BC isn't necessarily perpendicular. Hmm.Let me confirm: AD ⊥ CD, so angle at D is 90 degrees. So, ABCD is a right trapezoid with right angle at D. That simplifies things a bit. So, AD is vertical, DC is horizontal, AB is parallel to DC, so AB is also horizontal. Then, BC is the other leg, but not necessarily vertical.Given that M is the midpoint of CD. So, if CD is the base, then M divides CD into two equal parts. Then, AC is perpendicular to BM. Also, BC is perpendicular to BD. We need to find AB/CD.Maybe coordinate geometry would help here. Let's assign coordinates to the points. Let's place point D at the origin (0,0) since AD is perpendicular to CD. Then, since AD is vertical, point A would be at (0, h) where h is the height of the trapezoid. CD is horizontal, so point C is at (c, 0), and since AB is parallel to DC, point B would be at (b, h). Point M is the midpoint of CD, so coordinates of M would be ((c/2), 0).Wait, but hold on. If D is at (0,0), and CD is horizontal, then C is at (k, 0) for some k. Then, since AB is parallel to DC, and AD is vertical from D(0,0) to A(0, h). So point A is (0, h). Then, AB is horizontal from A(0, h) to B(b, h), so B is (b, h). Then, point C is (c, 0), so DC is from (0,0) to (c,0). Then, AB is from (0, h) to (b, h), so AB has length b, and DC has length c. So, the ratio AB/CD is b/c. So, our goal is to find b/c given the conditions.Now, the conditions: AC is perpendicular to BM, and BC is perpendicular to BD. Let me write these conditions in terms of coordinates.First, coordinates:A: (0, h)B: (b, h)C: (c, 0)D: (0, 0)M: midpoint of CD, so ((c + 0)/2, (0 + 0)/2) = (c/2, 0)First condition: AC ⊥ BM.Vector AC is from A(0, h) to C(c, 0), so the vector is (c, -h).Vector BM is from B(b, h) to M(c/2, 0), so the vector is (c/2 - b, -h).For two vectors to be perpendicular, their dot product is zero. So:(c)(c/2 - b) + (-h)(-h) = 0Simplify:c(c/2 - b) + h² = 0Which is:c²/2 - bc + h² = 0 --- Equation (1)Second condition: BC ⊥ BD.Vector BC is from B(b, h) to C(c, 0), vector is (c - b, -h)Vector BD is from B(b, h) to D(0, 0), vector is (-b, -h)Their dot product must be zero:(c - b)(-b) + (-h)(-h) = 0Simplify:- b(c - b) + h² = 0Which is:- bc + b² + h² = 0 --- Equation (2)So now we have two equations:1) c²/2 - bc + h² = 02) - bc + b² + h² = 0Let me write them again:Equation (1): (c²)/2 - bc + h² = 0Equation (2): b² - bc + h² = 0If we subtract Equation (2) from Equation (1):(c²/2 - bc + h²) - (b² - bc + h²) = 0 - 0Simplify:c²/2 - bc + h² - b² + bc - h² = 0The -bc and +bc cancel, h² - h² cancels, so:c²/2 - b² = 0So,c² = 2b²Which implies c = b√2Since lengths are positive, c = b√2.Okay, so c = b√2. Let's keep that in mind.Now, let's substitute c = b√2 into Equation (2):b² - bc + h² = 0But c = b√2, so:b² - b*(b√2) + h² = 0Simplify:b² - b²√2 + h² = 0Factor out b²:b²(1 - √2) + h² = 0Then,h² = b²(√2 - 1)So,h = b√(√2 - 1)Hmm, that's an expression for h in terms of b.Alternatively, maybe we can express h in terms of c. Since c = b√2, then b = c/√2. Then:h² = (c/√2)²(√2 - 1) = (c²/2)(√2 - 1)So,h² = (c²/2)(√2 - 1)But maybe we can also find h from Equation (1). Let's check.Equation (1):c²/2 - bc + h² = 0We already know c = b√2, so substitute:(b√2)^2 /2 - b*(b√2) + h² = 0Which is:(2b²)/2 - b²√2 + h² = 0Simplify:b² - b²√2 + h² = 0Which is the same as Equation (2), so that's consistent.So, h² = b²(√2 - 1)So h = b√(√2 - 1). Let me compute √(√2 - 1). Maybe rationalize it or express it differently.Note that √(√2 - 1) can be written as (sqrt(2) - 1)^(1/2). Alternatively, maybe express it in terms of trigonometric functions? Not sure if necessary. Let's see.Alternatively, let's see if we can find another relation.Wait, but maybe we can relate h and c. Since h² = (c²/2)(√2 - 1), so h = c * sqrt( (√2 - 1)/2 )Alternatively, sqrt( (√2 - 1)/2 ). Let me compute this:sqrt( (√2 - 1)/2 )Multiply numerator and denominator inside the sqrt by √2 + 1:sqrt( [ (√2 - 1)(√2 + 1) ] / [ 2(√2 + 1) ] )Compute numerator: (√2)^2 - 1 = 2 - 1 = 1So,sqrt( 1 / [ 2(√2 + 1) ] ) = 1 / sqrt( 2(√2 + 1) )But 2(√2 + 1) = 2√2 + 2. Hmm, not sure if helpful.Alternatively, rationalize the denominator:1 / sqrt(2(√2 + 1)) ) = sqrt(2(√2 + 1)) / [2(√2 + 1)]But this seems more complicated. Maybe just leave it as h = c * sqrt( (√2 - 1)/2 )But perhaps we can proceed without needing to compute h explicitly.Wait, but let me recall that in a trapezoid, the height is h, which is the distance between the two bases AB and DC. Since AD is perpendicular to DC, the height is AD, so AD = h. So, if we need any other lengths, perhaps?But maybe we can use similar triangles or other properties. Let me think.Alternatively, since we have coordinates, maybe compute the ratio AB/CD, which is b/c. Since we found c = b√2, so AB/CD = b / (b√2) = 1/√2. But wait, is that correct? Wait, but we need to confirm if this is the case, but maybe the answer is 1/√2. But let me check.Wait, but hold on. Wait, if AB is length b, since AB is from (0, h) to (b, h), so length is |b - 0| = b. CD is from (0,0) to (c,0), so length is c. Therefore, AB/CD is b/c = 1/√2. But is that the answer? Wait, but the problem gives additional conditions, so we need to check if this holds.But according to the equations we derived, c = b√2, so ratio is 1/√2. But let me check if the problem is correct. Let me verify with the given conditions.But hold on, if AB/CD is 1/√2, then rationalizing, that's √2/2, which is approximately 0.707. But let me check with the given perpendicular conditions. Maybe I made a mistake in calculation.Wait, let's re-express the equations step by step again.We had two equations:1) c²/2 - bc + h² = 02) b² - bc + h² = 0Subtracting equation 2 from equation 1 gives c²/2 - b² = 0, so c² = 2b², hence c = b√2. Therefore, AB/CD = b/c = 1/√2. However, I need to confirm if this is correct given the other condition.But let's also check the value of h. From equation 2: h² = bc - b². Since c = b√2, h² = b*(b√2) - b² = b²√2 - b² = b²(√2 - 1). So h = b√(√2 - 1). So, h is positive.But let's check if all conditions are satisfied.First, AC ⊥ BM. Let's compute vectors.AC is from A(0, h) to C(c, 0) = (c, -h)BM is from B(b, h) to M(c/2, 0) = (c/2 - b, -h)Dot product: c*(c/2 - b) + (-h)*(-h) = c²/2 - bc + h². According to equation 1, this is zero. So, condition satisfied.Second condition: BC ⊥ BD.Vector BC is from B(b, h) to C(c, 0) = (c - b, -h)Vector BD is from B(b, h) to D(0, 0) = (-b, -h)Dot product: (c - b)(-b) + (-h)(-h) = -bc + b² + h². According to equation 2, this is zero. So, condition satisfied.Therefore, the ratio AB/CD = 1/√2. But maybe the problem expects a rationalized form, which would be √2/2, but sometimes they prefer 1/√2. Let me check the problem statement. It says "put your final answer within boxed{}". Typically, in competition problems, rationalized denominators are preferred. So, √2/2 is more likely. But let me confirm.Alternatively, perhaps there is a miscalculation here. Wait, but according to the equations, AB/CD is indeed 1/√2. But maybe there's a step I missed.Wait, but let me think again. If AB is the upper base, then AB is from (0, h) to (b, h), so its length is |b - 0| = b. CD is from (0,0) to (c, 0), length c. So, ratio is b/c. Since we found c = b√2, then ratio is 1/√2. So, yes, that seems correct.But I need to make sure that there isn't another condition that affects this ratio. Wait, but we used both given perpendicular conditions (AC ⊥ BM and BC ⊥ BD) to derive the equations. Therefore, the ratio 1/√2 is the unique solution satisfying both conditions. Therefore, the answer should be 1/√2, which is equivalent to √2/2. So, writing it as boxed{dfrac{sqrt{2}}{2}}.But wait, hold on. Wait, let me check if this answer makes sense with the height. Suppose AB = 1, CD = √2, then h = √( (√2 - 1)/2 ) * √2 = ?Wait, perhaps not. Maybe I should verify with actual numbers.Let me take b = 1. Then c = √2. Then h² = (1)^2*(√2 - 1) = √2 - 1. So h = sqrt(√2 - 1). Let me compute sqrt(√2 - 1):sqrt(√2 - 1) ≈ sqrt(1.4142 - 1) ≈ sqrt(0.4142) ≈ 0.6428.So, points:A(0, 0.6428)B(1, 0.6428)C(1.4142, 0)D(0, 0)M(0.7071, 0)Now, check if AC is perpendicular to BM.Vector AC: from A(0, 0.6428) to C(1.4142, 0): (1.4142, -0.6428)Vector BM: from B(1, 0.6428) to M(0.7071, 0): (-0.2929, -0.6428)Dot product:(1.4142)(-0.2929) + (-0.6428)(-0.6428) ≈ (-0.4142) + (0.4132) ≈ -0.001Hmm, that's approximately zero, considering rounding errors. So, it's correct.Similarly, check BC ⊥ BD.Vector BC: from B(1, 0.6428) to C(1.4142, 0): (0.4142, -0.6428)Vector BD: from B(1, 0.6428) to D(0,0): (-1, -0.6428)Dot product:(0.4142)(-1) + (-0.6428)(-0.6428) ≈ -0.4142 + 0.4132 ≈ -0.001Again, approximately zero. So, the conditions are satisfied. Therefore, the ratio AB/CD = 1/√2 is correct.But let me check if the problem states AB is the top base or the bottom base. The problem says "Let ABCD be a trapezoid with AB parallel to DC." So, AB and DC are the two bases. Typically, in trapezoid notation, consecutive vertices are connected, so it's likely AB and DC are the two bases with AD and BC as legs. Since AD is perpendicular to DC, then AD is a leg, and DC is the base.But in terms of orientation, AB is parallel to DC, so if DC is the lower base, AB is the upper base. So, the ratio AB/CD is 1/√2. Therefore, the answer is boxed{dfrac{sqrt{2}}{2}}.Wait, but in the problem statement, it's possible that AB and DC could be the other way around, but since AB is parallel to DC, and AD is perpendicular to DC, it's standard to have DC as the base where the perpendicular leg is. So, the answer should be correct as 1/√2 or √2/2. But in the calculation above, with AB = 1 and CD = √2, the ratio is 1/√2, which is equivalent to √2/2. Wait, actually, 1/√2 is equal to √2/2. So, both are the same. Therefore, writing it as √2/2 is probably preferable.But let me check once again.Wait, AB is the upper base, length b. CD is the lower base, length c. Then, AB/CD = b/c. Since c = b√2, then AB/CD = 1/√2. Which is the same as √2/2. So, both forms are acceptable, but in competitions, usually rationalized forms are preferred, so boxed{dfrac{sqrt{2}}{2}}.Alternatively, if the answer is 1/√2, but I think √2/2 is more standard. Let me check the problem again. The problem says "find AB/CD". So, as a ratio. So, if AB is 1 and CD is √2, then AB/CD is 1/√2, which is equal to √2/2. So, either is acceptable, but perhaps the answer is simplified to √2/2. Let me confirm with calculation.Yes, 1/√2 = √2/2. So, both are correct, but in the problem proposed by Nathan Ramesh, maybe the answer is expected to be √2/2. So, I'll go with that.Final Answerboxed{dfrac{sqrt{2}}{2}}</think>Let (ABCD) be a trapezoid with (AB parallel DC). (M) is the midpoint of (CD), (AD perp CD), (AC perp BM), and (BC perp BD). We need to find (frac{AB}{CD}).1. Coordinate Setup: - Place (D) at ((0, 0)), (C) at ((c, 0)), (A) at ((0, h)), and (B) at ((b, h)). - Midpoint (M) of (CD) is (left(frac{c}{2}, 0right)).2. Vectors and Perpendicular Conditions: - Condition 1: (AC perp BM) - Vector (AC = (c, -h)) - Vector (BM = left(frac{c}{2} - b, -hright)) - Dot product: (c left(frac{c}{2} - bright) + (-h)(-h) = 0) - Simplifies to: (frac{c^2}{2} - bc + h^2 = 0) - Condition 2: (BC perp BD) - Vector (BC = (c - b, -h)) - Vector (BD = (-b, -h)) - Dot product: ((c - b)(-b) + (-h)(-h) = 0) - Simplifies to: (-bc + b^2 + h^2 = 0)3. Solving Equations: - From the equations (frac{c^2}{2} - bc + h^2 = 0) and (-bc + b^2 + h^2 = 0): - Subtract the second equation from the first: (frac{c^2}{2} - b^2 = 0) - This gives (c^2 = 2b^2) or (c = bsqrt{2}).4. Finding the Ratio: - The ratio (frac{AB}{CD}) is (frac{b}{c}). - Substituting (c = bsqrt{2}) gives (frac{b}{bsqrt{2}} = frac{1}{sqrt{2}} = frac{sqrt{2}}{2}).Thus, the final answer is (boxed{dfrac{sqrt{2}}{2}}).,<think>Okay, so I need to solve this trapezoid problem. Let me read it again. We have a trapezoid ABCD with AB parallel to DC. M is the midpoint of CD. There are a few conditions: AD is perpendicular to CD, AC is perpendicular to BM, and BC is perpendicular to BD. We need to find the ratio AB/CD.Hmm. Let me visualize this. Since AB is parallel to DC, and AD is perpendicular to CD, that means AD is also perpendicular to AB, right? Because if a line is perpendicular to one of the parallel sides, it's perpendicular to the other too. So AD is perpendicular to both AB and CD. Therefore, AD must be the height of the trapezoid.Let me try sketching this in my mind. Let's have DC as the base, horizontal. Then AD is vertical, going up from D to A. AB is parallel to DC, so AB is also horizontal. Then BC connects B to C. M is the midpoint of CD. So BM is a line from B to the midpoint of CD. Also, AC is perpendicular to BM. Additionally, BC is perpendicular to BD. That seems like a lot of orthogonality conditions. Maybe coordinate geometry will help here. Let's assign coordinates to the points.Let me place point D at the origin (0,0) since AD is perpendicular to CD, which makes calculations easier. Then, since DC is the base, let's let DC be along the x-axis. Let me denote the length of DC as 'c', so point C would be at (c, 0). Since M is the midpoint of CD, M would be at (c/2, 0).AD is perpendicular to CD, so AD is vertical. Let's denote the length of AD as 'h', so point A is at (0, h). AB is parallel to DC, so AB is horizontal. Let me denote the length of AB as 'k'. Then, point B must be at (k, h). Now, we need to figure out the coordinates of all points.So to recap, coordinates:- D: (0, 0)- C: (c, 0)- M: (c/2, 0)- A: (0, h)- B: (k, h)Now, we need to use the other conditions: AC is perpendicular to BM, and BC is perpendicular to BD.First, let's write down the coordinates:Point A: (0, h)Point C: (c, 0)So, vector AC is (c - 0, 0 - h) = (c, -h)Similarly, point B: (k, h)Point M: (c/2, 0)So, vector BM is (c/2 - k, 0 - h) = (c/2 - k, -h)Since AC is perpendicular to BM, their dot product should be zero. Let's compute that:AC ⋅ BM = (c)(c/2 - k) + (-h)(-h) = c(c/2 - k) + h² = (c²/2 - ck) + h² = 0So that's one equation: c²/2 - ck + h² = 0. Let's keep that in mind.Next condition: BC is perpendicular to BD.First, find vectors BC and BD.Point B: (k, h)Point C: (c, 0)Vector BC = (c - k, -h)Point B: (k, h)Point D: (0, 0)Vector BD = (-k, -h)Their dot product should be zero:BC ⋅ BD = (c - k)(-k) + (-h)(-h) = -k(c - k) + h² = -ck + k² + h² = 0So another equation: -ck + k² + h² = 0So now we have two equations:1) c²/2 - ck + h² = 02) -ck + k² + h² = 0Let me write them again:1) (c²)/2 - ck + h² = 02) k² - ck + h² = 0Interesting. Let's subtract equation 2 from equation 1:[ (c²)/2 - ck + h² ] - [ k² - ck + h² ] = 0 - 0Simplify:(c²)/2 - ck + h² - k² + ck - h² = 0Simplify terms:(c²)/2 - k² = 0So (c²)/2 = k² => c² = 2k² => c = k√2Since lengths are positive, we can take c = k√2.So we have c = k√2. Let's substitute this into one of the equations, say equation 2:k² - ck + h² = 0Substitute c = k√2:k² - k*(k√2) + h² = 0 => k² - k²√2 + h² = 0Therefore, h² = k²√2 - k² = k²(√2 - 1)So h = k√(√2 - 1)Hmm, that's a bit messy. Let's see if we can simplify √(√2 - 1). Let me compute √(√2 - 1):Let me denote x = √(√2 - 1). Then x² = √2 - 1. Let's square both sides again: x⁴ = (√2 - 1)² = 2 - 2√2 + 1 = 3 - 2√2. Not sure if that helps. Maybe not necessary to simplify further. Let's keep h as k√(√2 - 1) for now.Now, the problem asks for AB/CD. AB is the length of AB, which is k (since AB is from (0, h) to (k, h), so difference in x is k). CD is the length of DC, which is c, since DC is from (0,0) to (c,0). So AB/CD = k/c. But we found that c = k√2, so k/c = 1/√2. But 1/√2 is equal to √2/2. Wait, but hold on, let me confirm.Wait, AB is from (0, h) to (k, h), so yes, the length is k. CD is from (0,0) to (c,0), so length c. So the ratio AB/CD is k/c. Since c = k√2, ratio is k/(k√2) = 1/√2 = √2/2. But that would be the answer. But wait, did we use all the conditions? Wait, but we have h expressed in terms of k as h = k√(√2 - 1). But does this affect the ratio AB/CD? Wait, AB is k, CD is c = k√2, so the ratio is 1/√2 regardless of h. But the problem didn't give any specific lengths, so maybe the ratio is fixed? But the answer would be √2/2. But wait, the problem might have different conditions. Wait, maybe I made a mistake here.Wait, let me check. We had two equations:1) c²/2 - ck + h² = 02) k² - ck + h² = 0Subtracting 2 from 1 gives c²/2 - k² = 0 => c = k√2Then substituting back into equation 2 gives h² = k²√2 - k² = k²(√2 - 1)So h = k√(√2 - 1). So h is determined in terms of k, but AB and CD are k and c = k√2, so their ratio is 1/√2. But 1/√2 is equal to √2/2. Wait, but maybe I need to rationalize the denominator, so √2/2 is the simplified form.But let me check if this is correct. Let me verify with the conditions. Let's take c = k√2 and h = k√(√2 - 1). Let's check if AC is indeed perpendicular to BM.Coordinates:A: (0, h)C: (c, 0) = (k√2, 0)So vector AC is (k√2, -h) = (k√2, -k√(√2 - 1))Point B: (k, h)Point M: midpoint of CD, which is (c/2, 0) = (k√2/2, 0)So vector BM is (k√2/2 - k, 0 - h) = (k(√2/2 - 1), -h) = (k( (√2 - 2)/2 ), -k√(√2 - 1))Dot product of AC and BM:(k√2)(k( (√2 - 2)/2 )) + (-k√(√2 - 1))(-k√(√2 - 1))First term: k²√2*( (√2 - 2)/2 ) = k²*( (√2*(√2 - 2))/2 ) = k²*( (2 - 2√2)/2 ) = k²*(1 - √2)Second term: k²(√(√2 - 1))² = k²(√2 - 1)Total dot product: k²(1 - √2) + k²(√2 - 1) = k²(1 - √2 + √2 - 1) = k²(0) = 0. Good, so that's satisfied.Similarly, check BC perpendicular to BD.Point B: (k, h)Point C: (k√2, 0)Vector BC: (k√2 - k, -h) = (k(√2 - 1), -k√(√2 - 1))Point D: (0, 0)Vector BD: (-k, -h) = (-k, -k√(√2 - 1))Dot product BC ⋅ BD:(k(√2 - 1))*(-k) + (-k√(√2 - 1))*(-k√(√2 - 1))First term: -k²(√2 - 1)Second term: k²(√(√2 - 1))² = k²(√2 - 1)Total dot product: -k²(√2 - 1) + k²(√2 - 1) = 0. Perfect.So all conditions are satisfied. Therefore, AB/CD = k/c = 1/√2 = √2/2.But wait, the problem says "find AB/CD". So the answer is √2/2. But let me confirm once again. Wait, is AB the top base and CD the bottom base? Yes, since AB is parallel to DC, and AD is perpendicular to DC, so AD is the leg. So in a trapezoid, the two parallel sides are the bases. So AB and DC are the two bases. Since AB is the top one and DC is the bottom one. So AB/CD is k/c = 1/√2, which is equal to √2/2. So the ratio is √2/2.But let me check if there's another way this could be approached. Maybe using similar triangles or properties of trapezoids. Let me think.Alternatively, since M is the midpoint of CD, and AC is perpendicular to BM, maybe there's a geometric interpretation here. Also, BC is perpendicular to BD, which might imply some cyclic quadrilaterals or something. Hmm. But since we've already solved it using coordinates and verified the answer satisfies all conditions, maybe that's sufficient.But just to be thorough, let me consider if there's a way to approach this without coordinates.Given that ABCD is a trapezoid with AB || DC, AD ⊥ DC, so AD is the height. Let me denote AD = h, DC = c, AB = k. Then the coordinates approach was correct.Given that M is the midpoint of DC, so DM = MC = c/2.AC ⊥ BM: So in the trapezoid, diagonals and midpoints. Maybe using vectors or slopes.Alternatively, in coordinate terms, the slopes of AC and BM should be negative reciprocals.Slope of AC: (0 - h)/(c - 0) = -h/cSlope of BM: (0 - h)/(c/2 - k) = -h/(c/2 - k)If they are perpendicular, then (-h/c) * (-h/(c/2 - k)) = -1Wait, no. Wait, the product of slopes should be -1 for perpendicular lines. So:(-h/c) * (-h/(c/2 - k)) = -1So (h²)/(c(c/2 - k)) = -1But h² is positive, denominator is c(c/2 - k). For this product to be -1, denominator must be negative. So c(c/2 - k) < 0.So h² = -c(c/2 - k)But h² is positive, so -c(c/2 - k) must be positive. Thus, c(c/2 - k) < 0. Since c is positive (it's a length), then (c/2 - k) < 0 => c/2 < k => k > c/2.So h² = -c(c/2 - k) = c(k - c/2)Similarly, the other condition is BC perpendicular to BD. Let me check the slopes here.Slope of BC: (0 - h)/(c - k) = -h/(c - k)Slope of BD: (0 - h)/(0 - k) = -h/(-k) = h/kTheir product should be -1:(-h/(c - k)) * (h/k) = -1So (-h²)/(k(c - k)) = -1Multiply both sides by k(c - k):-h² = -k(c - k)So h² = k(c - k)From the previous condition, we have h² = c(k - c/2). So:c(k - c/2) = k(c - k)Let's expand both sides:Left: ck - (c²)/2Right: kc - k²Set equal:ck - (c²)/2 = ck - k²Subtract ck from both sides:- (c²)/2 = -k²Multiply both sides by -1:c²/2 = k² => c² = 2k² => c = k√2Which is the same as before. Then substitute back:h² = k(c - k) = k(k√2 - k) = k²(√2 - 1)So h = k√(√2 -1 )Same result. Therefore, ratio AB/CD = k/c = 1/√2 = √2/2.Therefore, the answer should be √2/2, which is equal to boxed{dfrac{sqrt{2}}{2}}.Wait, but let me check once more. In the problem statement, it says M is the midpoint of CD. We used that in our coordinate system, so M is at (c/2, 0). Then, AC is perpendicular to BM. The other condition is BC perpendicular to BD. All these were used in our equations, leading us to the ratio. So unless there's an error in the algebra, which I checked, the answer should be correct.Alternatively, maybe the answer is 1/2. Wait, but that would be if c = 2k. But according to our calculation, c = k√2. So the ratio is 1/√2, which is approximately 0.707. Whereas 1/2 is 0.5. But according to the problem conditions, with the perpendicularly conditions, I think √2/2 is correct.Wait, maybe I need to rationalize the answer. But √2/2 is already rationalized. So the answer is √2 over 2. So boxed, it's boxed{dfrac{sqrt{2}}{2}}.I think that's correct. Let me see if any step had an error. Wait, when we subtracted equation 2 from equation 1, we had:Equation1: (c²)/2 - ck + h² = 0Equation2: k² - ck + h² = 0Subtracting 2 from 1:(c²)/2 - ck + h² - k² + ck - h² = (c²)/2 - k² = 0 => c² = 2k². That's correct.Then substitution into equation2: k² - ck + h² = 0With c = k√2, then:k² - k*(k√2) + h² = k² - k²√2 + h² = 0 => h² = k²(√2 -1 )Yes. So no mistake here. Then ratio AB/CD is k/c = 1/√2 = √2/2. Correct.Therefore, the answer is boxed{dfrac{sqrt{2}}{2}}.Final Answerboxed{dfrac{sqrt{2}}{2}}</think>Given a trapezoid (ABCD) with (AB parallel DC), (M) as the midpoint of (CD), (AD perp CD), (AC perp BM), and (BC perp BD), we need to find the ratio (frac{AB}{CD}).1. Coordinate Assignment: - Place (D) at ((0, 0)), (C) at ((c, 0)), (A) at ((0, h)), (B) at ((k, h)), and (M) at ((frac{c}{2}, 0)).2. Vectors and Dot Products: - Vector (AC) is ((c, -h)). - Vector (BM) is ((frac{c}{2} - k, -h)). - Dot product of (AC) and (BM) must be zero: [ c left(frac{c}{2} - kright) + (-h)(-h) = 0 implies frac{c^2}{2} - ck + h^2 = 0 ]3. Perpendicular Conditions: - Vector (BC) is ((c - k, -h)). - Vector (BD) is ((-k, -h)). - Dot product of (BC) and (BD) must be zero: [ (c - k)(-k) + (-h)(-h) = 0 implies -ck + k^2 + h^2 = 0 ]4. Solving Equations: - From the two equations: [ frac{c^2}{2} - ck + h^2 = 0 quad text{and} quad k^2 - ck + h^2 = 0 ] - Subtracting the second from the first: [ frac{c^2}{2} - k^2 = 0 implies c^2 = 2k^2 implies c = ksqrt{2} ] - Substituting (c = ksqrt{2}) into the second equation: [ k^2 - k(ksqrt{2}) + h^2 = 0 implies h^2 = k^2(sqrt{2} - 1) ]5. Ratio Calculation: - The ratio (frac{AB}{CD}) is (frac{k}{c} = frac{k}{ksqrt{2}} = frac{1}{sqrt{2}} = frac{sqrt{2}}{2}).Thus, the final answer is (boxed{dfrac{sqrt{2}}{2}}).]

🔑:1. Identify the given conditions and setup the problem: - ABCD is a trapezoid with AB parallel DC. - M is the midpoint of CD. - AD perp CD, AC perp BM, and BC perp BD. - We need to find frac{AB}{CD}.2. Establish the similarity of triangles: - Let overline{BM} cap overline{AC} = E. - By the given conditions, triangle DBC sim triangle BAD and triangle CEM sim triangle AEB sim triangle CAD.3. Determine the properties of point M: - Since CM = DM and angle CBD is right, M is the circumcenter of triangle CBD. - This implies BM = DM = CM.4. Introduce angles and their relationships: - Let angle ABD = angle BDC = theta and angle ACD = angle BAC = lambda. - We have angle CEM = frac{pi}{2}, so angle EMC = frac{pi}{2} - lambda and angle EMD = frac{pi}{2} + lambda. - Since BM = DM, angle DBM = theta.5. Solve for theta and lambda: - From the angle relationships, theta = frac{pi}{2} - theta - lambda. - This simplifies to 2theta + lambda = frac{pi}{2}.6. Relate AB and CD using trigonometric identities: - AB tan theta = AD = CD tan lambda. - Therefore, frac{AB}{CD} = frac{tan lambda}{tan theta}.7. Express lambda in terms of theta: - From 2theta + lambda = frac{pi}{2}, we get lambda = frac{pi}{2} - 2theta. - Thus, frac{AB}{CD} = frac{tan (frac{pi}{2} - 2theta)}{tan theta}.8. Simplify using trigonometric identities: - tan (frac{pi}{2} - 2theta) = cot (2theta). - Therefore, frac{AB}{CD} = frac{cot (2theta)}{tan theta}.9. Use the double-angle identity for tangent: - cot (2theta) = frac{1}{tan (2theta)}. - tan (2theta) = frac{2 tan theta}{1 - tan^2 theta}. - Thus, cot (2theta) = frac{1 - tan^2 theta}{2 tan theta}.10. Combine the expressions: - frac{AB}{CD} = frac{frac{1 - tan^2 theta}{2 tan theta}}{tan theta} = frac{1 - tan^2 theta}{2 tan^2 theta}.11. Relate cos theta to frac{AB}{CD}: - cos theta = frac{AD}{BD} = frac{BD}{CD}. - cos^2 theta = frac{AB}{CD}.12. Solve the resulting equation: - cos^2 theta = frac{tan (frac{pi}{2} - 2theta)}{tan theta}. - tan (frac{pi}{2} - 2theta) = cot (2theta) = cos theta sin theta. - 2 cot (2theta) = sin (2theta). - 2 cos (2theta) = sin^2 (2theta). - 2 cos (2theta) = 1 - cos^2 (2theta). - cos^2 (2theta) + 2 cos (2theta) - 1 = 0. - cos (2theta) = sqrt{2} - 1. - sqrt{2} - 1 = 2 cos^2 theta - 1. - cos^2 theta = frac{sqrt{2}}{2}.The final answer is boxed{frac{sqrt{2}}{2}}.

🧠:[<think>Alright, let me try to figure out the range of this data set. The numbers given are 2, -2, 6, 4, -1. Hmm, okay. First, I remember that the range is a measure of how spread out the numbers are. To find the range, I think I need to subtract the smallest number in the set from the largest one. Let me make sure that's right.So, the formula for range is maximum value minus minimum value. Yeah, that sounds familiar from what I've studied before. So, I need to identify the highest and the lowest numbers here. Let me list them again to check: 2, -2, 6, 4, -1. Looking at these numbers, the first thing I notice is that there are both positive and negative values. The negative numbers might be the smallest ones here. Let's pick out the maximum first. Scanning through the numbers: 2 is positive, -2 is negative, 6 is positive, 4 is positive, and -1 is negative. So, the largest number here is 6, right? Because 6 is bigger than 4, 2, and all the others. Now, for the minimum. The negative numbers are -2 and -1. Between those, -2 is smaller than -1 because it's further left on the number line. So, the smallest number in the set is -2. Therefore, the range should be the maximum (6) minus the minimum (-2). Let's compute that: 6 - (-2). Subtracting a negative is like adding the positive, so that becomes 6 + 2. Which equals 8. Wait, let me double-check to make sure I didn't mix up any numbers. The data set is 2, -2, 6, 4, -1. So, sorted in order from smallest to largest would be: -2, -1, 2, 4, 6. Yep, that's right. So the minimum is definitely -2 and the maximum is 6. Calculating the difference again: 6 - (-2) = 6 + 2 = 8. That seems correct. I don't see any other numbers that could be higher or lower. So, the range is 8. I guess that's it. It's pretty straightforward once you identify the max and min. Just need to be careful with the negative numbers to make sure you don't mix up which one is smaller. But yeah, -2 is definitely less than -1. So subtracting a negative turns into addition. Just to recap: maximum value is 6, minimum is -2, 6 minus (-2) equals 8. The range of the data set is 8. I think that's the answer. Let me check one more time. If all the numbers were plotted on a number line, the distance from -2 to 6 would be 8 units. So, yeah, that's the range. Okay, confident now.</think>The range of a data set is calculated by subtracting the minimum value from the maximum value. Given the data set: (2), (-2), (6), (4), (-1)1. Identify the maximum value: (6)2. Identify the minimum value: (-2)3. Calculate the range: (6 - (-2) = 6 + 2 = 8)The range of this data set is (boxed{8}).,<think>Okay, so I need to find the range of this data set: 2, -2, 6, 4, -1. Hmm, let's see. First, what exactly is the range? I remember from my math class that the range is the difference between the highest and lowest values in a data set. So, basically, you take the maximum number and subtract the minimum number from it. That gives the range. Let me make sure I got that right. Yeah, I think that's it. Let me check with an example. If I have numbers 1, 3, 5, the range should be 5 - 1 = 4. That seems correct.Alright, applying that to the given data set. The numbers are 2, -2, 6, 4, -1. First, I need to identify the maximum and minimum values here. Let me list them again: 2, -2, 6, 4, -1. To find the maximum, I can just go through each number. Starting with the first one, 2. Then the next is -2, which is smaller. Then 6, which is bigger than 2. Then 4, which is less than 6. Then -1, which is still less than 6. So the maximum here is 6.Now for the minimum. Let's go through the numbers again. 2 is the first one. Next is -2, which is smaller. Then 6, which is not. Then 4, no. Then -1, which is bigger than -2. So the minimum is -2.So the range would be maximum minus minimum, which is 6 - (-2). Wait, subtracting a negative. That becomes addition, right? So 6 + 2 = 8. Therefore, the range should be 8.Let me double-check my steps. Found the maximum correctly: 6. Minimum: -2. Subtraction: 6 - (-2) = 8. Yeah, that seems right. Is there any chance I missed a number? Let me check again. The data set: 2, -2, 6, 4, -1. All numbers accounted for. Max is definitely 6, and min is -2. The calculation is straightforward. Hmm. I don't see any errors here.Wait a second, just to make sure, maybe I should sort the numbers from smallest to largest. That might help visualize. Let's sort them: -2, -1, 2, 4, 6. Yeah, that order makes sense. The smallest is -2 and the largest is 6. So the difference is 6 - (-2) = 8. Yep, that confirms it again. So the range is 8.I think that's solid. No mistakes here. So the answer should be 8.Final AnswerThe range of the data set is boxed{8}.</think>To find the range of the data set (2), (-2), (6), (4), (-1), we need to determine the difference between the maximum and minimum values.1. Identify the maximum value in the data set. The values are (2), (-2), (6), (4), (-1). The maximum value is (6).2. Identify the minimum value in the data set. The minimum value is (-2).3. Calculate the range by subtracting the minimum value from the maximum value: [ 6 - (-2) = 6 + 2 = 8 ]Thus, the range of the data set is (boxed{8}).]

🔑:To find the range of a set of data, we subtract the smallest number in the set from the largest number. For the given set of data 2, -2, 6, 4, -1, we identify the smallest and largest numbers:- The smallest number is -2.- The largest number is 6.Therefore, the range is calculated as follows:[text{Range} = text{Largest number} - text{Smallest number} = 6 - (-2) = 6 + 2 = 8.]Thus, the range of this data set is boxed{8}.

🧠:[<think>Okay, let's see here. The problem is about pioneers in a group, some of whom won medals and others got certificates. The total number of pioneers is 33, and out of these, 8 won medals. The question is asking how many received certificates instead.Hmm, so if there are 33 pioneers in total and 8 of them got medals, then the rest must have gotten certificates. So maybe I just need to subtract the number of medal winners from the total number to find out how many got certificates. Let me check that logic again.Total pioneers = 33. Medal winners = 8. So certificates would be 33 minus 8. Let me do that calculation. 33 minus 8 is... 25. Wait, is that right? 33 take away 8: 33 - 8. Let me count. 33 minus 10 is 23, but since it's only 8, that's 2 more, so 23 + 2 = 25. Yeah, that seems correct.But hold on, the problem mentions that the medals were for first, second, or third place. Does that affect anything? Like, maybe there are 3 medals per event or something? But the problem doesn't specify how many events there were. It just says 8 pioneers won medals because they took first, second, or third place. So maybe each of those 8 pioneers got a medal in some event, but regardless of how many events, the total medalists are 8. So the number of certificate recipients would still be total minus medalists, which is 33 - 8 = 25.I think that's all there is to it. The mention of first, second, third might just be explaining why they got medals, but since the total number of medalists is given as 8, we don't need to break it down further. So the answer should be 25 pioneers received certificates.</think>To determine the number of pioneers who received certificates, subtract the number of medal winners from the total number of pioneers: [33 text{ total pioneers} - 8 text{ medal winners} = 25 text{ certificate recipients}.]The answer is boxed{25}.,<think>Okay, let's see. There are 33 pioneers in total. Out of these, 8 won medals because they got first, second, or third places. The rest got certificates for participation. The question is asking how many received certificates.Hmm, so if there are 33 total and 8 of them got medals, then the ones who got certificates must be the total minus the ones with medals. So that would be 33 minus 8. Let me check that again. Yeah, because all the others besides the medal winners get certificates. So 33 minus 8 equals... 25. Wait, 33 minus 10 is 23, so minus 8 would be 25. Yeah, that's right. So 25 pioneers received certificates. I think that's it. Let me make sure there's no trick or anything. The problem says 8 pioneers were able to win medals because they took first, second, or third place. So each of those positions is a different medal, but does that mean there were 8 medals or 8 pioneers? The wording says 8 pioneers were able to win medals. So regardless of how many medals there are, 8 people got medals. Therefore, subtract 8 from 33 to get 25. Yeah, that seems straightforward. I don't think there's anything else to consider here. So the answer should be 25.</think>To determine the number of pioneers who received certificates, subtract the number of medal winners from the total number of pioneers:[33 text{ total pioneers} - 8 text{ medal winners} = 25 text{ certificate recipients}]The number of pioneers in the group who received certificates is boxed{25}.]

🔑:25 Pioneers of this group received certificates.

🧠:[<think>Okay, so I need to solve the equation 1/x = 2/(x + 1). Hmm, let's see. Fractional equations can sometimes be tricky because of the variables in the denominators. I remember that one way to solve such equations is by cross-multiplication. Let me try that first.Cross-multiplication means multiplying the numerator of one fraction by the denominator of the other fraction and setting them equal. So, if I have 1/x = 2/(x + 1), then cross-multiplying would give me 1*(x + 1) = 2*x. Let me write that out:1*(x + 1) = 2*xSimplifying both sides, the left side becomes x + 1 and the right side is 2x. So:x + 1 = 2xHmm, now I need to solve for x. Let me subtract x from both sides to get the x terms on one side. That would be:x + 1 - x = 2x - xSimplifying both sides, the left side becomes 1 and the right side is x. So:1 = xSo, x equals 1. Wait, but I should check if this solution is valid. Sometimes when we solve equations with variables in the denominators, we might end up with a solution that makes the denominator zero, which is undefined. Let's plug x = 1 back into the original equation to check.Original equation: 1/x = 2/(x + 1)Substituting x = 1:Left side: 1/1 = 1Right side: 2/(1 + 1) = 2/2 = 1So both sides equal 1. That works. Also, checking the denominators with x = 1: the denominators are x = 1 and x + 1 = 2, neither of which are zero. So x = 1 is a valid solution.Wait, is there another way this could be approached? Maybe by finding a common denominator? Let me try that method as well to confirm.Starting with 1/x = 2/(x + 1). Let's subtract 2/(x + 1) from both sides to set the equation to zero:1/x - 2/(x + 1) = 0To combine these fractions, I need a common denominator. The denominators are x and x + 1, so the common denominator would be x(x + 1). Let's rewrite each fraction with that denominator:[1*(x + 1)] / [x(x + 1)] - [2*x] / [x(x + 1)] = 0Combine the numerators:[(x + 1) - 2x] / [x(x + 1)] = 0Simplify the numerator:x + 1 - 2x = -x + 1So the equation becomes:(-x + 1) / [x(x + 1)] = 0A fraction is equal to zero when its numerator is zero (provided the denominator isn't zero). So set the numerator equal to zero:-x + 1 = 0Solving for x:- x + 1 = 0 => -x = -1 => x = 1Same result as before. So x = 1 is the solution. Again, checking the denominator x(x + 1) when x = 1: 1*(1 + 1) = 2 ≠ 0, so no issues here. Therefore, x = 1 is indeed the valid solution.Is there a possibility I missed another solution? Let me think. The equation is 1/x = 2/(x + 1). After cross-multiplying, I got a linear equation, which only has one solution. So unless there was an extraneous solution introduced, which we checked and didn't happen here, there's only one solution. So I think x = 1 is the only answer.Alternatively, maybe graphing both sides of the equation. The left side is y = 1/x, which is a hyperbola with a vertical asymptote at x = 0. The right side is y = 2/(x + 1), which is a hyperbola with a vertical asymptote at x = -1. The solution to the equation is where these two hyperbolas intersect. Since both are hyperbolas, they might intersect at one point. From the algebra, we found x = 1. If I were to graph them, when x = 1, y = 1 for both, so the point (1, 1) is the intersection. That seems reasonable.Another check: let's pick a value greater than 1, say x = 2. Left side: 1/2 = 0.5. Right side: 2/(2 + 1) = 2/3 ≈ 0.666. Not equal. What about x approaching infinity? 1/x approaches 0, and 2/(x + 1) also approaches 0. But they both approach zero, but maybe at different rates. However, since they are both approaching zero, the equation might not have another solution at infinity.If x is negative, let's say x = -0.5. Then left side: 1/(-0.5) = -2. Right side: 2/(-0.5 + 1) = 2/(0.5) = 4. Not equal. How about x = -2? Left side: 1/(-2) = -0.5. Right side: 2/(-2 + 1) = 2/(-1) = -2. Not equal. So between x = -1 and x = 0, maybe? Let's try x approaching -1 from the right. x = -0.9. Left side: 1/(-0.9) ≈ -1.111. Right side: 2/(-0.9 + 1) = 2/0.1 = 20. Not close. If x approaches 0 from the left, left side approaches negative infinity, right side approaches 2/(0 + 1) = 2. So they won't intersect there. So indeed, the only intersection is at x = 1.Therefore, after checking via multiple methods: cross-multiplication, common denominators, graphical intuition, and testing values, the solution x = 1 is valid and the only solution.Final AnswerThe solution is boxed{1}.</think>To solve the fractional equation (frac{1}{x} = frac{2}{x+1}), we start by cross-multiplying to eliminate the denominators:[1 cdot (x + 1) = 2 cdot x]Simplifying both sides, we get:[x + 1 = 2x]Subtracting (x) from both sides:[1 = x]We check the solution (x = 1) by substituting it back into the original equation. The left side becomes (frac{1}{1} = 1) and the right side becomes (frac{2}{1 + 1} = 1), confirming both sides are equal. Additionally, the denominators (x = 1) and (x + 1 = 2) are not zero, so the solution is valid.To further verify, we can combine the fractions over a common denominator:[frac{1}{x} - frac{2}{x+1} = 0]Rewriting with the common denominator (x(x + 1)):[frac{(x + 1) - 2x}{x(x + 1)} = 0]Simplifying the numerator:[frac{-x + 1}{x(x + 1)} = 0]Setting the numerator equal to zero:[-x + 1 = 0 implies x = 1]Both methods confirm that (x = 1) is the valid solution. Therefore, the solution to the equation is (boxed{1}).,<think>Okay, so I need to solve the equation 1/x equals 2/(x + 1). Hmm, fractional equation. Alright, let's see. First, maybe I can cross-multiply to get rid of the denominators. That usually works for proportions like this. So, if I cross-multiply, that would mean multiplying both sides by x and by (x + 1), right? Let me write that down.Starting with 1/x = 2/(x + 1). If I cross-multiply, I get 1*(x + 1) = 2*x. Let me check that. Yeah, cross-multiplication is multiplying the numerator of each side by the denominator of the other side. So, 1 times (x + 1) equals 2 times x. Okay, that seems right.So expanding that, 1*(x + 1) is just x + 1, and 2*x is 2x. So now the equation is x + 1 = 2x. Hmm, solving for x. Let me subtract x from both sides to get the variables on one side. If I subtract x from both sides, that leaves me with 1 = 2x - x, which simplifies to 1 = x. So x equals 1?Wait, but before I get too confident, I should check if this solution is valid. Because sometimes when you have variables in the denominator, you might end up with a solution that makes the original denominators zero, which is undefined. So let me substitute x = 1 back into the original equation to verify.Original equation: 1/x = 2/(x + 1). Plugging in x = 1, that becomes 1/1 = 2/(1 + 1). Simplifying both sides: 1 = 2/2, which is 1 = 1. Yup, that works. So x = 1 is a valid solution. But wait, just to make sure I didn't make any mistakes in cross-multiplying. Let me go through the steps again. The original equation is 1/x = 2/(x + 1). Cross-multiplying gives 1*(x + 1) = 2*x. So x + 1 = 2x. Subtract x: 1 = x. Yep, same result. And plugging back in works. Is there another solution? Let's see. When I subtracted x from both sides, the equation became 1 = x. So that's the only solution. If I had gotten a quadratic equation, there might be more, but this is linear. So only x = 1. Alternatively, maybe I can solve the equation by getting all terms to one side. Let me try that. Starting with 1/x = 2/(x + 1). Subtract 2/(x + 1) from both sides: 1/x - 2/(x + 1) = 0. To combine these fractions, I need a common denominator, which would be x(x + 1). So:[ (1*(x + 1) - 2*x) ] / [x(x + 1)] = 0.Simplifying the numerator: (x + 1 - 2x) = (-x + 1). So the equation becomes (-x + 1)/[x(x + 1)] = 0. A fraction is zero when the numerator is zero (as long as the denominator isn't zero). So set -x + 1 = 0. Solving that gives x = 1. Then check the denominator x(x + 1) when x = 1: 1*(1 + 1) = 2, which is not zero. So again, x = 1 is valid. Okay, that method also gives x = 1. So that must be the solution. Just to recap: cross-multiplying gives a straightforward way to solve the equation, leading to x = 1. Checking in the original equation confirms it's valid. Another method of subtracting the right side and combining fractions also leads to x = 1. Both approaches confirm the same answer. I don't think I missed anything here. The equation is straightforward. The key steps were cross-multiplying to eliminate denominators and then solving the resulting linear equation. Then, importantly, checking that the solution doesn't make any denominators zero in the original equation. Since x = 1 doesn't cause any issues (denominators are 1 and 2, both non-zero), it's a valid solution. Therefore, the solution is x = 1.Final AnswerThe solution is boxed{1}.</think>To solve the fractional equation (frac{1}{x} = frac{2}{x+1}), we start by cross-multiplying to eliminate the denominators. This gives us:[1 cdot (x + 1) = 2 cdot x]Expanding both sides, we get:[x + 1 = 2x]Subtracting (x) from both sides, we find:[1 = x]We check the solution by substituting (x = 1) back into the original equation:[frac{1}{1} = frac{2}{1 + 1} implies 1 = 1]This confirms that (x = 1) is a valid solution. Additionally, solving by combining fractions also leads to the same result. Therefore, the solution to the equation is:[boxed{1}]]

🔑:To solve the given fractional equation frac{1}{x}=frac{2}{x+1}, we follow these steps:1. Multiply both sides by x(x+1) to eliminate the denominators:[x(x+1) cdot frac{1}{x} = x(x+1) cdot frac{2}{x+1}]2. Simplify both sides:[x+1 = 2x]3. Solve for x:begin{align*}x+1 &= 2x 1 &= 2x - x 1 &= xend{align*}4. Verify that the solution does not make the original denominators equal to zero:[x(x+1) = 1(1+1) = 2 neq 0]Therefore, the solution to the original equation is boxed{x=1}.

🧠:[<think>Okay, let's see. Marie does three tasks one after another, and each takes the same amount of time. She starts the first task at 1:00 PM and finishes the second task at 2:40 PM. I need to figure out when she finishes the third task. The answer choices are from 3:10 PM to 4:30 PM, so I need to calculate the time each task takes and add that to the finish time of the second task.First, let's figure out how much time she spent on the first two tasks. She started the first task at 1:00 PM and finished the second task at 2:40 PM. So from 1:00 PM to 2:40 PM is the time taken for two tasks. Let me calculate the duration between 1:00 PM and 2:40 PM.From 1:00 PM to 2:00 PM is 1 hour. Then from 2:00 PM to 2:40 PM is 40 minutes. So total time is 1 hour and 40 minutes. That's the time she spent on two tasks. Since all tasks are equally time-consuming, each task takes the same amount of time. So I need to divide 1 hour and 40 minutes by 2 to get the time per task.Let me convert that into minutes to make it easier. 1 hour is 60 minutes, so 60 + 40 = 100 minutes. So two tasks take 100 minutes. Therefore, each task takes 100 minutes divided by 2, which is 50 minutes per task.Wait, so each task is 50 minutes long. Then, she does three tasks in a row. The first task starts at 1:00 PM. Let's check if that makes sense. First task: 1:00 PM to 1:50 PM. Then the second task would be from 1:50 PM to 2:40 PM. Wait, that's exactly when she finishes the second task. So yes, that matches the given information. So each task is 50 minutes.Therefore, the third task would start at 2:40 PM and take another 50 minutes. So we need to add 50 minutes to 2:40 PM to find out when she finishes the third task.Adding 50 minutes to 2:40 PM. Let's break it down. 2:40 PM plus 20 minutes is 3:00 PM. Then we have 30 minutes left (since 50 - 20 = 30). So 3:00 PM plus 30 minutes is 3:30 PM. So 2:40 PM + 50 minutes = 3:30 PM. Therefore, she finishes the third task at 3:30 PM.Looking at the answer choices, option B is 3:30 PM. So I think that's the correct answer. Let me just verify once again.She starts the first task at 1:00 PM. Each task is 50 minutes. So first task ends at 1:50 PM. Second task starts at 1:50 PM and ends at 2:40 PM, which matches the problem statement. Then the third task starts at 2:40 PM, adds 50 minutes, which is 3:30 PM. Yeah, that seems right. So the answer should be B, 3:30 PM.Final Answerboxed{B}</think>Marie starts the first task at 1:00 PM and finishes the second task at 2:40 PM. The duration from 1:00 PM to 2:40 PM is 1 hour and 40 minutes, which is equivalent to 100 minutes. Since she completes two tasks in this time, each task takes ( frac{100}{2} = 50 ) minutes. To find when she finishes the third task, we add 50 minutes to the time she finishes the second task (2:40 PM). Adding 50 minutes to 2:40 PM:- 2:40 PM + 20 minutes = 3:00 PM- 3:00 PM + 30 minutes = 3:30 PMThus, Marie finishes the third task at 3:30 PM. The answer is (boxed{B}).,<think>Okay, so Marie is doing three tasks one after another, and each task takes the same amount of time. She starts the first task at 1:00 PM and finishes the second task at 2:40 PM. The question is asking when she finishes the third task. The answer choices are from 3:10 PM to 4:30 PM in 20 to 30-minute increments. Let me think through this step by step.First, let's parse the timeline here. She starts the first task at 1:00 PM. Since the tasks are done in a row without breaks, the end time of one task is the start time of the next. She finishes the second task at 2:40 PM. So, from the start of the first task to the end of the second task, how much time has passed?Starting at 1:00 PM, ending at 2:40 PM. That's 1 hour and 40 minutes. Wait, 1:00 PM to 2:00 PM is 1 hour, and then from 2:00 PM to 2:40 PM is 40 minutes. So total time is 1 hour 40 minutes, which is 100 minutes.Now, during this period, she has completed two tasks. Since the tasks are equally time-consuming, each task must take 100 minutes divided by 2, which is 50 minutes per task. Wait, hold on: if she started the first task at 1:00 PM, then the first task would take 50 minutes, ending at 1:50 PM. Then she starts the second task immediately at 1:50 PM, which would take another 50 minutes, ending at 2:40 PM. That checks out with the given information. So each task is 50 minutes long.Therefore, the third task would start immediately after the second task ends at 2:40 PM. Adding 50 minutes to 2:40 PM. Let's calculate that. 2:40 PM plus 50 minutes. Let's break it down: 2:40 PM plus 20 minutes gets us to 3:00 PM. Then we have 50 - 20 = 30 minutes left. So 3:00 PM plus 30 minutes is 3:30 PM. So the third task should end at 3:30 PM. Let me confirm that again.Start of first task: 1:00 PM. First task ends at 1:50 PM. Second task starts at 1:50 PM, ends at 2:40 PM. Third task starts at 2:40 PM, ends at 2:40 PM + 50 minutes. 50 minutes after 2:40 PM. Let's add 50 minutes to 2:40 PM. 2:40 PM plus 50 minutes: 2:40 + 20 = 3:00 PM, then 30 minutes more is 3:30 PM. So that seems right.Looking at the answer choices, option B is 3:30 PM, which would be the correct answer. Let me double-check to make sure I didn't make a mistake in my time calculations. Sometimes with time, it's easy to mix up the hours and minutes.So total duration from 1:00 PM to 2:40 PM is 1 hour 40 minutes, which is 100 minutes. She completes two tasks in that time, so each task is 50 minutes. Therefore, the third task, starting at 2:40 PM, will take 50 minutes. 2:40 PM plus 50 minutes: 40 minutes gets us to 3:20 PM, then adding the remaining 10 minutes would be 3:30 PM. Wait, actually, 50 minutes from 2:40 PM. Let me count it step by step. From 2:40 PM, adding 20 minutes is 3:00 PM. Then 30 minutes more is 3:30 PM. So yes, 2:40 PM + 50 minutes = 3:30 PM. That's correct.Alternatively, maybe I can check by converting the times to minutes past a certain hour. For example, 1:00 PM is 13:00 in 24-hour time. Let's convert all times to minutes since midnight for clarity. 1:00 PM is 13 hours, which is 13 * 60 = 780 minutes. 2:40 PM is 14 hours and 40 minutes, which is 14*60 + 40 = 840 + 40 = 880 minutes. The total duration between starting the first task and finishing the second is 880 - 780 = 100 minutes. Two tasks take 100 minutes, so each task is 50 minutes. Therefore, the third task starts at 880 minutes (2:40 PM) and ends at 880 + 50 = 930 minutes. Converting 930 minutes back to hours: 930 divided by 60 is 15.5 hours, which is 3:30 PM. So that's 15 hours (3 PM) plus 30 minutes. Therefore, 3:30 PM. So that's another way to confirm.Alternatively, let's think about how many minutes each task is. If she started at 1:00 PM, and each task is 50 minutes, then first task ends at 1:50 PM. Second task ends at 1:50 PM + 50 minutes = 2:40 PM, which matches the given information. Then third task starts at 2:40 PM, ends at 2:40 PM + 50 minutes = 3:30 PM. So yes, answer B.But let me check again because sometimes when problems mention "finishes the second task at 2:40 PM," does that mean the duration from start to finish is two tasks? Yes. Because she starts the first task at 1:00 PM, does the first task, then the second task. So the time from 1:00 PM to 2:40 PM is the duration of two tasks, which is 100 minutes. So each task is 50 minutes. Then the third task takes another 50 minutes. So starting at 2:40 PM, adding 50 minutes, you get to 3:30 PM. So answer B.Alternatively, maybe someone might think that the time from 1:00 PM to 2:40 PM is 1 hour 40 minutes, which is 100 minutes, but perhaps divide by three tasks? Wait, no, the problem says she does three tasks in a row. She starts the first at 1:00 PM and finishes the second at 2:40 PM, so the first two tasks take 100 minutes. Therefore, each task is 50 minutes. So third task starts at 2:40 PM, ends at 3:30 PM.Alternatively, could there be a mistake in the problem interpretation? For example, maybe when it says "finishes the second task at 2:40 PM," does that mean she started the second task at 1:00 PM? But that wouldn't make sense. No, the first task starts at 1:00 PM, then second task starts after the first, third after the second. So the first task ends at 1:00 PM + task duration. Then second task ends at 1:00 PM + 2*task duration, which is 2:40 PM, so 2*task duration is 100 minutes, hence task duration is 50 minutes. Therefore, third task ends at 2:40 PM + 50 minutes = 3:30 PM. Therefore, answer B.Alternatively, maybe some people might miscalculate the duration between 1:00 PM and 2:40 PM. Let's verify that. From 1:00 PM to 2:00 PM is 1 hour, which is 60 minutes. From 2:00 PM to 2:40 PM is 40 minutes. So total 60 + 40 = 100 minutes. So two tasks take 100 minutes. So each task is 50 minutes. Therefore, the third task will take another 50 minutes. Starting at 2:40 PM, adding 50 minutes: 2:40 PM + 50 minutes. Breaking into hours and minutes. 50 minutes is 10 minutes short of an hour. So adding 50 minutes to 2:40 PM: 2 hours + 40 minutes + 50 minutes = 2 hours + 90 minutes. 90 minutes is 1 hour and 30 minutes, so total 3 hours and 30 minutes. Wait, starting from 1:00 PM? Wait, no. Starting at 2:40 PM, adding 50 minutes. 2:40 PM plus 50 minutes. Let's think in terms of adding minutes to the time.2:40 PM plus 20 minutes is 3:00 PM. Then, remaining 30 minutes (since 50 - 20 = 30). So 3:00 PM + 30 minutes is 3:30 PM. So yes, that's correct.Alternatively, in another approach, 2:40 PM plus 50 minutes. If we consider 2:40 PM as 2 hours and 40 minutes, adding 50 minutes would be 2 hours + (40 + 50) minutes = 2 hours + 90 minutes. 90 minutes is 1 hour and 30 minutes. Therefore, 2 hours + 1 hour + 30 minutes = 3 hours 30 minutes. So 3:30 PM. That's another way to see it.Therefore, the correct answer is 3:30 PM, which is option B. So I think that's the right choice.Final Answerboxed{B}</think>Marie starts the first task at 1:00 PM and finishes the second task at 2:40 PM. The total time taken to complete the first two tasks is from 1:00 PM to 2:40 PM, which is 1 hour and 40 minutes (or 100 minutes). Since the tasks are equally time-consuming, each task takes ( frac{100}{2} = 50 ) minutes.To find the finish time of the third task, we add 50 minutes to the finish time of the second task (2:40 PM). Adding 50 minutes to 2:40 PM:- 2:40 PM + 20 minutes = 3:00 PM- 3:00 PM + 30 minutes = 3:30 PMThus, Marie finishes the third task at 3:30 PM.[boxed{B}]]

🔑:Marie finishes 2 tasks in 1 hour and 40 minutes. Therefore, one task should take 50 minutes to finish. 50 minutes after 2!:!40 PM is 3!:!30 PM, so our answer is boxed{textbf{(B) }text{3:30 PM}}

🧠:[<think>Alright, so I need to figure out the range of values for the real number ( a ) such that the function ( f(x) = e^x - a ) has odd symmetry points. Hmm, let's start by understanding what odd symmetry points are. The problem says that if there exists an ( x_0 neq 0 ) such that ( f(x_0) = -f(-x_0) ), then the points ( (x_0, f(x_0)) ) and ( (-x_0, f(-x_0)) ) form a pair of odd symmetry points. Okay, so for such points to exist, there must be some non-zero ( x_0 ) where the function evaluated at ( x_0 ) is the negative of the function evaluated at ( -x_0 ). That is, ( f(x_0) = -f(-x_0) ). Let me write that down with the given function.Given ( f(x) = e^x - a ), substituting into the condition gives:( e^{x_0} - a = -left(e^{-x_0} - aright) )Simplify the right side:( e^{x_0} - a = -e^{-x_0} + a )Now, let's bring all terms to one side:( e^{x_0} + e^{-x_0} - a - a = 0 )Which simplifies to:( e^{x_0} + e^{-x_0} - 2a = 0 )So, ( e^{x_0} + e^{-x_0} = 2a )Hmm, the left side of the equation is ( e^{x_0} + e^{-x_0} ), which is a well-known expression. That's equal to ( 2cosh(x_0) ), where ( cosh ) is the hyperbolic cosine function. So, substituting that in:( 2cosh(x_0) = 2a )Divide both sides by 2:( cosh(x_0) = a )So, the equation we need to solve is ( cosh(x_0) = a ), where ( x_0 neq 0 ).But what is the range of ( cosh(x) )? Remember that the hyperbolic cosine function ( cosh(x) ) is defined as ( frac{e^x + e^{-x}}{2} ), and it's an even function. Its minimum value occurs at ( x = 0 ), where ( cosh(0) = 1 ). For ( x neq 0 ), ( cosh(x) ) is always greater than 1. So, ( cosh(x) geq 1 ) for all real ( x ), and it's equal to 1 only when ( x = 0 ).But in our case, ( x_0 neq 0 ), so ( cosh(x_0) > 1 ). Therefore, for the equation ( cosh(x_0) = a ) to have a solution ( x_0 neq 0 ), ( a ) must be greater than 1. Thus, the range of values for ( a ) is ( a > 1 ). Therefore, ( a ) must be in the interval ( (1, infty) ).Wait, let me double-check. The problem states that there exists an ( x_0 neq 0 ) such that ( f(x_0) = -f(-x_0) ). So, solving the equation leads to ( cosh(x_0) = a ). Since ( cosh(x_0) ) can take any value from 1 to infinity, but ( x_0 neq 0 ), so ( cosh(x_0) ) can't be 1. Therefore, ( a ) must be greater than 1. Yes, that seems right. Therefore, the range of ( a ) is ( a > 1 ), which in interval notation is ( (1, infty) ).Final AnswerThe range of values for the real number ( a ) is (boxed{(1, +infty)}).</think>Given the function ( f(x) = e^x - a ), we need to determine the range of values for ( a ) such that the function has odd symmetry points. A pair of odd symmetry points exists if there is an ( x_0 neq 0 ) such that ( f(x_0) = -f(-x_0) ).Starting with the given condition:[f(x_0) = -f(-x_0)]Substituting ( f(x) = e^x - a ) into the equation, we get:[e^{x_0} - a = -left(e^{-x_0} - aright)]Simplifying the right-hand side:[e^{x_0} - a = -e^{-x_0} + a]Rearranging terms:[e^{x_0} + e^{-x_0} - 2a = 0]This simplifies to:[e^{x_0} + e^{-x_0} = 2a]Recognizing that ( e^{x_0} + e^{-x_0} = 2cosh(x_0) ), we substitute this in:[2cosh(x_0) = 2a]Dividing both sides by 2:[cosh(x_0) = a]The hyperbolic cosine function ( cosh(x) ) has a minimum value of 1 at ( x = 0 ) and is greater than 1 for all ( x neq 0 ). Since ( x_0 neq 0 ), ( cosh(x_0) > 1 ). Therefore, ( a ) must be greater than 1.The range of values for the real number ( a ) is (boxed{(1, +infty)}).,<think>Okay, so I need to find the range of values for the real number ( a ) such that the function ( f(x) = e^x - a ) has odd symmetry points. First, let me make sure I understand what odd symmetry points are. The problem says that if there's an ( x_0 neq 0 ) where ( f(x_0) = -f(-x_0) ), then the points ( (x_0, f(x_0)) ) and ( (-x_0, f(-x_0)) ) form a pair of odd symmetry points. So essentially, for some non-zero ( x_0 ), the function evaluated at ( x_0 ) is the negative of the function evaluated at ( -x_0 ). That reminds me of odd functions, which satisfy ( f(-x) = -f(x) ) for all ( x ). But here, it's not required for all ( x ), just for some specific ( x_0 neq 0 ). So the function doesn't have to be odd overall, just needs to have at least one pair of points that exhibit this odd symmetry.Given ( f(x) = e^x - a ), I need to find the values of ( a ) for which there exists an ( x_0 neq 0 ) such that ( f(x_0) = -f(-x_0) ). Let me write that equation out:( e^{x_0} - a = -left( e^{-x_0} - a right) )Let me simplify this equation step by step. First, distribute the negative sign on the right-hand side:( e^{x_0} - a = -e^{-x_0} + a )Now, let's bring all terms to one side to combine like terms. Let's add ( e^{-x_0} ) to both sides and subtract ( a ) from both sides:( e^{x_0} + e^{-x_0} - a - a = 0 )Wait, actually, let me check that again. Starting from:( e^{x_0} - a = -e^{-x_0} + a )If I add ( e^{-x_0} ) to both sides and add ( a ) to both sides:Left side becomes ( e^{x_0} - a + e^{-x_0} + a = e^{x_0} + e^{-x_0} )Right side becomes ( -e^{-x_0} + a + e^{-x_0} + a = 2a )So the equation simplifies to:( e^{x_0} + e^{-x_0} = 2a )Hmm, okay. So ( 2a = e^{x_0} + e^{-x_0} ). Now, ( e^{x_0} + e^{-x_0} ) is a familiar expression. That's equal to ( 2cosh(x_0) ), where ( cosh(x) ) is the hyperbolic cosine function. The hyperbolic cosine function has a minimum value of 1 at ( x = 0 ) and increases as ( |x| ) increases. So, ( e^{x_0} + e^{-x_0} geq 2 ), with equality if and only if ( x_0 = 0 ). But in our case, ( x_0 neq 0 ), so ( e^{x_0} + e^{-x_0} > 2 ).Therefore, ( 2a > 2 implies a > 1 ).Wait, but let me verify this again. If ( x_0 neq 0 ), then ( e^{x_0} + e^{-x_0} > 2 ), so ( 2a = e^{x_0} + e^{-x_0} ) implies that ( a = frac{e^{x_0} + e^{-x_0}}{2} ). Since ( frac{e^{x_0} + e^{-x_0}}{2} = cosh(x_0) ), and ( cosh(x_0) > 1 ) for all ( x_0 neq 0 ), then ( a > 1 ).But hold on, is there a restriction on ( a ) beyond this? Let's see. The equation ( e^{x} + e^{-x} = 2a ) must have a solution ( x neq 0 ). Since ( e^{x} + e^{-x} ) can take any value in ( [2, infty) ), then to have a solution, ( 2a ) must be in ( (2, infty) ). Therefore, ( a > 1 ).But let me think again. If ( a = 1 ), then ( 2a = 2 ), which is the minimum value of ( e^{x} + e^{-x} ), achieved only at ( x = 0 ). But since we require ( x_0 neq 0 ), then ( a = 1 ) is not acceptable. So indeed, ( a ) must be greater than 1.Therefore, the range of values for ( a ) is ( (1, infty) ).Wait, but hold on a second. Let me check my algebra again because sometimes when solving equations, you might introduce extraneous solutions or miss some constraints. Let me go back to the original equation:( f(x_0) = -f(-x_0) )So:( e^{x_0} - a = - (e^{-x_0} - a ) )Simplify right-hand side:( -e^{-x_0} + a )So the equation is:( e^{x_0} - a = -e^{-x_0} + a )Bring all terms to left-hand side:( e^{x_0} - a + e^{-x_0} - a = 0 )Combine like terms:( e^{x_0} + e^{-x_0} - 2a = 0 )So:( e^{x_0} + e^{-x_0} = 2a )Which is the same as before. Therefore, we must have ( 2a = e^{x_0} + e^{-x_0} ). Since ( e^{x_0} + e^{-x_0} geq 2 ), with equality only at ( x_0 = 0 ), but we need ( x_0 neq 0 ), so ( e^{x_0} + e^{-x_0} > 2 ). Hence, ( 2a > 2 ) which implies ( a > 1 ). Therefore, the range of ( a ) is ( a > 1 ).But let me check with an example. Suppose ( a = 2 ). Then, the equation is ( e^{x} + e^{-x} = 4 ). Solving for ( x ):Let me set ( y = e^{x} ). Then ( e^{-x} = 1/y ), so the equation becomes:( y + 1/y = 4 )Multiply both sides by ( y ):( y^2 + 1 = 4y )Bring all terms to left:( y^2 - 4y + 1 = 0 )Quadratic in ( y ), solutions are:( y = [4 ± sqrt(16 - 4)] / 2 = [4 ± sqrt(12)] / 2 = [4 ± 2*sqrt(3)] / 2 = 2 ± sqrt(3) )Thus, ( y = 2 + sqrt(3) ) or ( y = 2 - sqrt(3) ). Since ( y = e^{x} > 0 ), both solutions are acceptable because ( 2 - sqrt(3) ) is approximately 2 - 1.732 = 0.267 > 0. Therefore, ( x = ln(2 + sqrt(3)) ) or ( x = ln(2 - sqrt(3)) ). But ( ln(2 - sqrt(3)) = ln(1/(2 + sqrt(3))) = -ln(2 + sqrt(3)) ). So indeed, if ( x_0 = ln(2 + sqrt(3)) ), then ( -x_0 = ln(2 - sqrt(3)) ). Therefore, these are valid solutions. So when ( a = 2 ), there exists such an ( x_0 neq 0 ). So that works.If I pick ( a = 1 ), the equation would be ( e^{x} + e^{-x} = 2 ), which is only true when ( x = 0 ), which is not allowed. So ( a = 1 ) is excluded.If I pick ( a = 0.5 ), then the equation ( e^{x} + e^{-x} = 1 ). But ( e^{x} + e^{-x} geq 2 ), so there's no solution, which is consistent with the conclusion that ( a > 1 ).Therefore, my conclusion seems correct. The range of ( a ) is all real numbers greater than 1.Final AnswerThe range of values for the real number ( a ) is (boxed{(1, +infty)}).</think>Given the function ( f(x) = e^x - a ), we need to find the range of values for ( a ) such that the function has odd symmetry points. This means there exists an ( x_0 neq 0 ) such that ( f(x_0) = -f(-x_0) ).Starting with the given condition:[f(x_0) = -f(-x_0)]we substitute the function into the equation:[e^{x_0} - a = -left( e^{-x_0} - a right)]Simplifying the right-hand side:[e^{x_0} - a = -e^{-x_0} + a]Rearranging terms to combine like terms:[e^{x_0} + e^{-x_0} - 2a = 0]This simplifies to:[e^{x_0} + e^{-x_0} = 2a]Recognizing that ( e^{x_0} + e^{-x_0} ) is the hyperbolic cosine function, we know it has a minimum value of 2 at ( x_0 = 0 ). Since ( x_0 neq 0 ), the value of ( e^{x_0} + e^{-x_0} ) must be greater than 2. Therefore:[2a > 2 implies a > 1]Thus, the range of values for the real number ( a ) is (boxed{(1, +infty)}).]

🔑:AnalysisThis problem examines the symmetry of functions. From the concept of "odd symmetry points" mentioned in the problem, we can derive the relationship between the coordinates of two points and simplify it to obtain the result.SolutionGiven that the function has odd symmetry points,there exist two points on the graph of the function that are symmetric about the origin.Let the two points be P(x_1, y_1), Q(x_2, y_2),then y_1 = e^{x_1} - a, y_2 = e^{x_2} - a,which leads to e^{x_1} - a = -(e^{x_2} - a),thus 2a = e^{x_1} + e^{x_2} geqslant 2 sqrt{e^{x_1} cdot e^{x_2}} = 2 sqrt{e^0} = 2,since x_1 neq x_2,we have a > 1.Therefore, the answer is boxed{(1, +infty)}.