🧠:[<think>Okay, let's see. The problem is asking about the area bounded by the parabola y = 2x², the vertical line x = 2, and the x-axis y = 0. Then there's a multiple-choice question about how the interval [0, 2] is divided into n equal parts, and identifying the i-th interval. Hmm, so first, maybe I should focus on the second part, the multiple-choice question, because the first part might be context or related? Wait, the way it's phrased: "When the interval [0, 2] is divided into n equal parts, the i-th interval is (　　)". So the options are A to D. Then, perhaps this is related to setting up a Riemann sum to calculate the area mentioned? Maybe the question is part of a larger problem where you need to set up an integral by dividing the interval [0, 2] into n parts, and then taking the limit as n approaches infinity. But the actual question here is just about identifying the correct subinterval when [0, 2] is divided into n equal parts. So the answer would be one of the options given. Let me think.First, the interval is [0, 2], divided into n equal parts. Each part would have a width of (2 - 0)/n = 2/n. So each subinterval has length 2/n. Now, the question is, for the i-th interval, what are its endpoints? Let's list them. The first interval (i=1) should start at 0 and end at 2/n. The second interval (i=2) starts at 2/n and ends at 4/n. Continuing this way, the i-th interval should start at (i - 1)*(2/n) and end at i*(2/n). So in mathematical terms, the i-th interval is [2(i - 1)/n, 2i/n]. Let me check the options. Option D is exactly that: [2(i - 1)/n, 2i/n]. So that should be the correct answer. Let me confirm by plugging in some values. For example, if n = 2, the interval [0, 2] is divided into two parts, each of width 1. So the first interval (i=1) should be [0, 1], and the second (i=2) [1, 2]. Using option D: For i=1, [2(1 - 1)/2, 2(1)/2] = [0, 1]. For i=2, [2(2 - 1)/2, 2*2/2] = [2(1)/2, 4/2] = [1, 2]. Perfect, that matches. Let's check another option just to be sure. Option A is [(i - 1)/n, i/n]. If n=2, then i=1 would be [0, 0.5], which is not the case. So that's incorrect. Similarly, option C: [2(i - 2)/n, 2(i -1)/n]. For i=1, that would be [2(-1)/2, 2(0)/2] = [-1, 0], which is outside the interval. So C is wrong. Option B: [i/n, (i +1)/n]. For i=1, [1/2, 2/2] = [0.5, 1], which is the second half of the first subinterval if we split into two. Not correct. So yes, D is the right answer.But wait, the problem mentions the area bounded by the parabola, x=2, and y=0. Maybe that's a red herring or maybe the question is part of a setup for computing that area using Riemann sums. Let me verify. The area under y=2x² from x=0 to x=2 is indeed the integral from 0 to 2 of 2x² dx, which is (2/3)x³ evaluated from 0 to 2, so (2/3)(8) - 0 = 16/3. But the question here is about dividing [0,2] into n equal parts. So when setting up a Riemann sum, typically you partition the interval into n subintervals, each of width Δx = 2/n. Then each subinterval is [x_{i-1}, x_i], where x_i = 0 + iΔx = 2i/n. Wait, but in the standard setup, x_i = a + iΔx, where a is the left endpoint. So if the interval is [0, 2], then x_0 = 0, x_1 = 2/n, x_2 = 4/n, ..., x_i = 2i/n. Therefore, the i-th subinterval is [x_{i-1}, x_i] = [2(i - 1)/n, 2i/n], which is exactly option D. So that's consistent. Therefore, the answer is D.But just to make sure, let me think again. Suppose we have n subintervals. Each has width 2/n. Starting at 0, the first subinterval is 0 to 2/n, which is [2(1 - 1)/n, 2(1)/n] = [0, 2/n]. Second is 2/n to 4/n, which is [2(2 - 1)/n, 2(2)/n] = [2/n, 4/n], and so on. So yes, each i-th interval is [2(i - 1)/n, 2i/n], which is option D. Therefore, the answer is D.Final Answerboxed{D}</think>To find the correct interval when the interval ([0, 2]) is divided into (n) equal parts, we start by noting that each subinterval will have a width of (frac{2}{n}). For the (i)-th interval, the starting point is calculated as ((i - 1) times frac{2}{n}) and the ending point is (i times frac{2}{n}). Therefore, the (i)-th interval is (left[ frac{2(i - 1)}{n}, frac{2i}{n} right]).To verify, consider dividing the interval ([0, 2]) into (n = 2) parts. The first interval ((i = 1)) should be ([0, 1]) and the second interval ((i = 2)) should be ([1, 2]). Using the formula for the (i)-th interval:- For (i = 1): (left[ frac{2(1 - 1)}{2}, frac{2 times 1}{2} right] = [0, 1])- For (i = 2): (left[ frac{2(2 - 1)}{2}, frac{2 times 2}{2} right] = [1, 2])This matches the expected intervals. Checking other options confirms they do not provide the correct intervals. Thus, the correct answer is (boxed{D}).,<think>Alright, so I need to figure out which interval corresponds to the i-th interval when the interval [0, 2] is divided into n equal parts. The options are A, B, C, D. Let me start by understanding the problem.First, the main task here is about partitioning the interval [0, 2] into n equal subintervals and identifying the correct expression for the i-th subinterval. The answer choices are different expressions for these intervals, so I need to recall how to divide an interval into equal parts.Let me remember that when you divide an interval [a, b] into n equal parts, each part has a length of (b - a)/n. In this case, the interval is from 0 to 2, so each subinterval should have a length of (2 - 0)/n = 2/n. That seems straightforward. So each subinterval is 2/n long.Now, the next thing is to figure out the starting and ending points of the i-th subinterval. Let's think about how intervals are divided. If we have n equal parts, then the points dividing the intervals would be at 0, 2/n, 4/n, 6/n, ..., up to 2n/n = 2. So the k-th point is at (2(k-1))/n for k from 1 to n+1? Wait, maybe I need to index this properly.Wait, actually, if we have n subintervals, there are n+1 partition points. The first partition point is at x = 0, then x = 2/n, x = 4/n, ..., up to x = 2. So the i-th subinterval would start at x = 2(i - 1)/n and end at x = 2i/n. Because for i = 1, the first interval is [0, 2/n], which is when i=1, 2(1-1)/n = 0 and 2(1)/n = 2/n. Similarly, for i=2, it would be [2/n, 4/n], and so on, until i=n, which would be [2(n-1)/n, 2]. That seems right.Let me check the options:A: [(i−1)/n, i/n]B: [i/n, (i+1)/n]C: [2(i−2)/n, 2(i−1)/n]D: [2(i−1)/n, 2i/n]Hmm. So according to my reasoning, the correct interval for the i-th subinterval is [2(i−1)/n, 2i/n], which is option D. Let me verify this.Suppose n=2. Then each subinterval should be length 2/2=1. So the intervals would be [0,1] and [1,2]. Let's check option D with i=1: [2(1−1)/2, 2(1)/2] = [0, 1]. For i=2: [2(2−1)/2, 2(2)/2] = [2(1)/2, 4/2] = [1, 2]. Perfect. That matches the expected intervals for n=2.What about option A? For n=2 and i=1: [(1−1)/2, 1/2] = [0, 0.5]. Then i=2: [1/2, 1]. But those intervals are of length 0.5, but we need length 1. So A is not correct.Option B: For i=1, [1/2, 2/2] = [0.5, 1]; i=2: [1, 1.5], but 1.5 is beyond the interval [0,2], so that's not right either.Option C: For i=1, [2(1−2)/2, 2(1−1)/2] = [2(-1)/2, 0] = [-1, 0]. But that's outside the original interval [0,2]. So C is invalid even for i=1. So definitely not.Therefore, the correct answer should be D.But wait, let me double-check with another example. Let's take n=3. Then each interval is length 2/3. The intervals should be [0, 2/3], [2/3, 4/3], [4/3, 2]. Let's check option D:i=1: [2(1−1)/3, 2(1)/3] = [0, 2/3]i=2: [2(2−1)/3, 2(2)/3] = [2/3, 4/3]i=3: [2(3−1)/3, 2(3)/3] = [4/3, 6/3] = [4/3, 2]Perfect. So that's correct. Option D gives the right intervals.Alternatively, if we had used option A for n=3, the first interval would be [0, 1/3], which is length 1/3, but we need length 2/3, so that's wrong. Similarly, option B would start at 1/3, which is also not correct. So, yes, D is the correct answer.Just to make sure I didn't make any mistake in reasoning. The original interval is from 0 to 2. Divided into n parts, each of width 2/n. The starting point of the i-th interval is (i-1)*(2/n), and the ending point is i*(2/n). Hence, written as [2(i-1)/n, 2i/n], which is exactly option D.Therefore, the answer should be D.Final Answerboxed{D}</think>To determine the correct interval for the i-th subinterval when the interval ([0, 2]) is divided into (n) equal parts, we start by noting that each subinterval has a length of (frac{2}{n}). The partition points dividing the interval ([0, 2]) are at (0, frac{2}{n}, frac{4}{n}, ldots, 2). The i-th subinterval starts at (frac{2(i-1)}{n}) and ends at (frac{2i}{n}). Verification with examples:- For (n = 2), the intervals should be ([0, 1]) and ([1, 2]). Using option D, the intervals are ([0, 1]) for (i = 1) and ([1, 2]) for (i = 2), which is correct.- For (n = 3), the intervals should be ([0, frac{2}{3}]), ([frac{2}{3}, frac{4}{3}]), and ([frac{4}{3}, 2]). Using option D, the intervals are correctly calculated as ([0, frac{2}{3}]), ([frac{2}{3}, frac{4}{3}]), and ([frac{4}{3}, 2]) for (i = 1, 2, 3) respectively.Thus, the correct answer is (boxed{D}).]

🔑:Solution: Dividing the interval [0, 2] into n equal parts, each interval has a length of frac{2}{n},Thus, the division points are x_0=0, x_1=0+ frac{2}{n}, …, x_n=2,Therefore, the i-th interval is left[ frac{2(i-1)}{n}, frac{2i}{n} right],Hence, the correct option is boxed{D}.According to the conditions, by dividing the interval equally and determining the length of each interval, the conclusion can be drawn.This question mainly examines the definition of integration. Using the definition of integration to divide the interval equally is the key to solving this problem, which is quite basic.

🧠:[<think>Okay, let's see. I need to find the minimum value of the function ( y = 2m cdot x + frac{3}{x - 1} + 1 ) where ( x > 1 ). First, they gave me that ( m = frac{tan 22.5^circ}{1 - tan^2 22.5^circ} ). Hmm, maybe I should start by simplifying that expression for ( m ). Let me recall some trigonometric identities. I know that the tangent of a double angle is ( tan 2theta = frac{2 tan theta}{1 - tan^2 theta} ). Comparing this with the expression for ( m ), it looks similar but not exactly the same. Let me check: if I let ( theta = 22.5^circ ), then ( 2theta = 45^circ ). So, ( tan 45^circ = frac{2 tan 22.5^circ}{1 - tan^2 22.5^circ} ). But ( tan 45^circ = 1 ), so:( 1 = frac{2 tan 22.5^circ}{1 - tan^2 22.5^circ} )Therefore, rearranging, ( frac{tan 22.5^circ}{1 - tan^2 22.5^circ} = frac{1}{2} ). Wait, that's exactly the expression for ( m ). So ( m = frac{1}{2} ). Oh, okay, that simplifies things. So then ( 2m = 2 times frac{1}{2} = 1 ). So the function simplifies to ( y = x + frac{3}{x - 1} + 1 ). Now, I need to find the minimum value of this function for ( x > 1 ). Since we have ( x ) and ( frac{3}{x - 1} ), this looks like a candidate for using calculus or the AM-GM inequality. Let me try both approaches to see which is easier.First, let's try calculus. To find the minimum, take the derivative of ( y ) with respect to ( x ) and set it to zero. ( y = x + frac{3}{x - 1} + 1 )Compute the derivative:( y' = 1 - frac{3}{(x - 1)^2} )Set ( y' = 0 ):( 1 - frac{3}{(x - 1)^2} = 0 )Solving for ( x ):( frac{3}{(x - 1)^2} = 1 )( (x - 1)^2 = 3 )Take square roots:( x - 1 = sqrt{3} ) or ( x - 1 = -sqrt{3} )But since ( x > 1 ), we discard the negative root:( x = 1 + sqrt{3} )Now, plug this back into the original function to find the minimum value:( y = (1 + sqrt{3}) + frac{3}{(1 + sqrt{3} - 1)} + 1 )Simplify:( y = 1 + sqrt{3} + frac{3}{sqrt{3}} + 1 )Simplify ( frac{3}{sqrt{3}} = sqrt{3} ):( y = 1 + sqrt{3} + sqrt{3} + 1 = 2 + 2sqrt{3} )So the minimum value is ( 2 + 2sqrt{3} ), which is option C.Alternatively, let me try using the AM-GM inequality. The function is ( y = x + frac{3}{x - 1} + 1 ). Let me make a substitution to make it easier. Let ( t = x - 1 ), so since ( x > 1 ), ( t > 0 ). Then ( x = t + 1 ), and substituting back:( y = (t + 1) + frac{3}{t} + 1 = t + 1 + frac{3}{t} + 1 = t + frac{3}{t} + 2 )So now, the problem reduces to minimizing ( t + frac{3}{t} ) for ( t > 0 ), and then adding 2. Let's focus on ( t + frac{3}{t} ).Using AM-GM inequality: For positive numbers ( a ) and ( b ), the arithmetic mean is greater than or equal to the geometric mean. So, ( frac{a + b}{2} geq sqrt{ab} ). Equality holds when ( a = b ).But in this case, we have ( t + frac{3}{t} ). To apply AM-GM, we can think of it as ( t + frac{3}{t} = t + frac{3}{t} ). Let me consider splitting the 3 into three 1s or something else. Wait, maybe not. Alternatively, think of it as ( t + frac{3}{t} = t + frac{3}{t} ), which is similar to ( a + frac{b}{a} ).Alternatively, perhaps set ( a = t ), ( b = frac{3}{t} ). Then, AM-GM gives ( frac{a + b}{2} geq sqrt{ab} ), so ( a + b geq 2sqrt{ab} ). Therefore,( t + frac{3}{t} geq 2 sqrt{ t times frac{3}{t} } = 2 sqrt{3} )Equality holds when ( t = frac{3}{t} implies t^2 = 3 implies t = sqrt{3} ).Therefore, the minimum of ( t + frac{3}{t} ) is ( 2sqrt{3} ), occurring at ( t = sqrt{3} ). Therefore, the minimum value of ( y = t + frac{3}{t} + 2 ) is ( 2sqrt{3} + 2 ), which is ( 2 + 2sqrt{3} ), same as before. So the answer is option C.Wait, but let me confirm once more. The substitution seems correct. ( x = t + 1 ), so when ( t = sqrt{3} ), ( x = 1 + sqrt{3} ), which matches the calculus result. Then substituting back into the original equation gives the same minimum value. Both methods agree. The answer must be C: ( 2 + 2sqrt{3} ).Final Answerboxed{C}</think>Given ( m = frac{tan 22.5^circ}{1 - tan^2 22.5^circ} ), we recognize this as related to the double-angle identity for tangent. Using the identity ( tan 2theta = frac{2 tan theta}{1 - tan^2 theta} ), we find that ( tan 45^circ = 1 = frac{2 tan 22.5^circ}{1 - tan^2 22.5^circ} ). Solving this, we get ( frac{tan 22.5^circ}{1 - tan^2 22.5^circ} = frac{1}{2} ), so ( m = frac{1}{2} ). Thus, ( 2m = 1 ).The function simplifies to ( y = x + frac{3}{x - 1} + 1 ) for ( x > 1 ). To find the minimum value, we use calculus. Taking the derivative:[y' = 1 - frac{3}{(x - 1)^2}]Setting ( y' = 0 ):[1 - frac{3}{(x - 1)^2} = 0 implies (x - 1)^2 = 3 implies x - 1 = sqrt{3} implies x = 1 + sqrt{3}]Substituting ( x = 1 + sqrt{3} ) back into the function:[y = (1 + sqrt{3}) + frac{3}{sqrt{3}} + 1 = 1 + sqrt{3} + sqrt{3} + 1 = 2 + 2sqrt{3}]Using the AM-GM inequality on ( t + frac{3}{t} ) (where ( t = x - 1 )):[t + frac{3}{t} geq 2sqrt{3}]Equality holds when ( t = sqrt{3} ), leading to ( y = 2sqrt{3} + 2 ).Thus, the minimum value of the function is ( 2 + 2sqrt{3} ), which corresponds to option C.[boxed{C}],<think>Alright, let's try to figure out this problem step by step. First, let me understand what's being asked here. We're given a value m, which is equal to tan(22.5°) divided by (1 - tan²22.5°). Then, we need to find the minimum value of the function y = 2m·x + 3/(x - 1) + 1 for x > 1. The options are given, so maybe after calculating m and then finding the minimum, one of those options will match.Let me start by calculating m. The expression is m = tan(22.5°) / (1 - tan²22.5°). Hmm, 22.5 degrees is half of 45 degrees, so maybe there's a trigonometric identity that can simplify this. Let me recall that the double angle formula for tangent is tan(2θ) = 2tanθ / (1 - tan²θ). Wait, that's similar to the denominator here. Let's see, if we set θ = 22.5°, then 2θ = 45°, so tan(45°) = 2tan(22.5°)/(1 - tan²22.5°). But tan(45°) is 1. So, 1 = 2tan(22.5°)/(1 - tan²22.5°). Therefore, (1 - tan²22.5°) = 2tan(22.5°). Therefore, m = tan(22.5°)/(1 - tan²22.5°) = tan(22.5°)/(2tan(22.5°)) = 1/2. Wait, that seems straightforward. So m is 1/2. Let me verify that again.Yes, using the double angle formula: tan(2θ) = 2tanθ / (1 - tan²θ). So if θ = 22.5°, then tan(45°) = 1 = 2tan(22.5°)/(1 - tan²22.5°). So rearranged, 1 - tan²22.5° = 2tan22.5°, so m = tan22.5° / (1 - tan²22.5°) = tan22.5° / (2tan22.5°) = 1/2. Yep, that makes sense. So m is 1/2. That simplifies things.So the function y becomes y = 2*(1/2)*x + 3/(x - 1) + 1. Simplifying that, 2*(1/2) is 1, so y = x + 3/(x - 1) + 1. Let me rewrite that as y = x + 1 + 3/(x - 1). Hmm. So we need to find the minimum value of this function for x > 1.Alright, to find the minimum of y = x + 1 + 3/(x - 1) where x > 1. Let me consider substitution to make it easier. Let t = x - 1. Since x > 1, t > 0. Then, x = t + 1. Substitute back into y: y = (t + 1) + 1 + 3/t = t + 1 + 1 + 3/t = t + 2 + 3/t. So the function simplifies to y = t + 3/t + 2, where t > 0.Now, finding the minimum of y = t + 3/t + 2. This looks like a standard application of the AM-GM inequality or calculus. Let's try both ways to confirm.First, using calculus. Take the derivative of y with respect to t: dy/dt = 1 - 3/t². Set derivative equal to zero: 1 - 3/t² = 0 => 1 = 3/t² => t² = 3 => t = sqrt(3) (since t > 0, we discard the negative root). Then, the minimum occurs at t = sqrt(3). Then, compute y at t = sqrt(3): y = sqrt(3) + 3/sqrt(3) + 2 = sqrt(3) + sqrt(3) + 2 = 2sqrt(3) + 2. Wait, but looking at the answer choices, option C is 2 + 2sqrt(3), which is the same as 2sqrt(3) + 2. So that seems like the answer. But hold on, let me check using AM-GM to see if that's correct.Using AM-GM inequality: For positive numbers a and b, the minimum of a + b is 2sqrt(ab). But here we have t + 3/t. Let me write t + 3/t = t + 3/(t) = t + 3*(1/t). So a = t and b = 3/t. Then, by AM-GM, t + 3/t >= 2*sqrt(t*(3/t)) = 2*sqrt(3). So the minimum of t + 3/t is 2sqrt(3), achieved when t = sqrt(3/t) => t^2 = 3 => t = sqrt(3), same as before. Then, adding 2 gives y >= 2sqrt(3) + 2. So the minimum value is 2 + 2sqrt(3), which is option C. Hmm, but let me check if that's correct.Wait, but maybe I made a mistake in substitution. Let's check the substitution again. Original function y = x + 1 + 3/(x - 1). Let t = x - 1, so x = t + 1. Then, substituting gives y = (t + 1) + 1 + 3/t = t + 2 + 3/t. So yes, that's correct. So y = t + 3/t + 2. Then, AM-GM gives t + 3/t >= 2sqrt(3). So y >= 2sqrt(3) + 2. So the minimum is 2 + 2sqrt(3). So answer C. But let me check with calculus again.Take derivative of y with respect to t: dy/dt = 1 - 3/t². Setting to zero gives t = sqrt(3). Then, substituting back: y = sqrt(3) + 3/sqrt(3) + 2. 3/sqrt(3) is sqrt(3)*sqrt(3)/sqrt(3) = sqrt(3). So sqrt(3) + sqrt(3) = 2sqrt(3). Then, plus 2 gives 2sqrt(3) + 2. So yes, same result. Therefore, the minimum value is 2 + 2sqrt(3), which is option C. So that's the answer. But let me check if the problem had any other trick or if I messed up in substitution.Wait, the original problem says x > 1. After substitution, t = x - 1, so t > 0, which is okay. The substitution is valid, and the calculus and AM-GM both confirm that the minimum is at t = sqrt(3). Then, converting back to x, x = t + 1 = sqrt(3) + 1. So x is indeed greater than 1. So that's okay. So the minimal value is 2 + 2sqrt(3), which is option C. Therefore, the answer should be C.But let me check the answer options again:A: 2B: 2√3C: 2 + 2√3D: 2√3 - 2Yes, so 2 + 2√3 is option C, which matches our result. Therefore, the correct answer is C.Final Answerboxed{C}</think>Given ( m = frac{tan 22.5^circ}{1 - tan^2 22.5^circ} ), we start by simplifying ( m ). Using the double angle formula for tangent, (tan(2theta) = frac{2tantheta}{1 - tan^2theta}), we set (theta = 22.5^circ). Then, (tan(45^circ) = 1 = frac{2tan(22.5^circ)}{1 - tan^2(22.5^circ)}), leading to (1 - tan^2(22.5^circ) = 2tan(22.5^circ)). Thus, ( m = frac{tan(22.5^circ)}{2tan(22.5^circ)} = frac{1}{2} ).Next, we substitute ( m = frac{1}{2} ) into the function ( y = 2m cdot x + frac{3}{x-1} + 1 ), resulting in ( y = x + frac{3}{x-1} + 1 ). We introduce the substitution ( t = x - 1 ) (where ( t > 0 ) since ( x > 1 )), transforming the function into ( y = (t + 1) + 1 + frac{3}{t} = t + 2 + frac{3}{t} ).To find the minimum value, we use the AM-GM inequality on ( t ) and ( frac{3}{t} ):[t + frac{3}{t} geq 2sqrt{t cdot frac{3}{t}} = 2sqrt{3}]The equality holds when ( t = sqrt{3} ). Substituting ( t = sqrt{3} ) back into the function, we get:[y = sqrt{3} + 2 + frac{3}{sqrt{3}} = sqrt{3} + 2 + sqrt{3} = 2sqrt{3} + 2]Thus, the minimum value of the function is ( 2 + 2sqrt{3} ), which corresponds to option C.[boxed{C}]]

🔑:Since x>1, we have x-1>0. We have m= frac {tan 22.5^circ }{1-tan^{2}22.5^circ }= frac {1}{2}tan 45^circ= frac {1}{2}, Thus, y=2mcdot x+ frac {3}{x-1}+1=x+ frac {3}{x-1}+1=(x-1)+ frac {3}{x-1}+2geq 2 sqrt {3}+2, Therefore, the correct choice is: boxed{text{C}}. By using the double angle formula to find m, and then applying the basic inequality, we can find the minimum value of the function y=2mcdot x+ frac {3}{x-1}+1 (x>1). This problem tests the minimum value of the function y=2mcdot x+ frac {3}{x-1}+1 (x>1), the application of the double angle formula, and the correct use of the basic inequality is key.

🧠:[<think>Okay, so I have this problem here about arithmetic sequences. Let me read it again: "Given that {a_n} is an arithmetic sequence with the first term being 2 and a non-zero common difference, if a₁, a₃, a₆ form a geometric sequence, then the sum of the first 9 terms of {a_n} is (　　) A: 26, B: 30, C: 36, D: 40."Alright, let's break this down. First, it's an arithmetic sequence, which means each term increases by a constant difference, right? The first term is 2, and the common difference is non-zero. So, the nth term of the arithmetic sequence can be written as a_n = a₁ + (n-1)d, where a₁ is 2, and d is the common difference. So, substituting the values, a_n = 2 + (n-1)d.Now, the problem says that a₁, a₃, a₆ form a geometric sequence. A geometric sequence is one where each term is multiplied by a constant ratio. So, in a geometric sequence, the square of the middle term is equal to the product of the first and third terms. That is, if the three terms are x, y, z, then y² = x*z.So, applying this to the terms a₁, a₃, a₆. Let's first write out what these terms are in the arithmetic sequence.Given that a₁ is 2, as given. Then a₃ is the third term of the arithmetic sequence. Since the nth term is a_n = 2 + (n-1)d, then a₃ = 2 + (3-1)d = 2 + 2d. Similarly, a₆ = 2 + (6-1)d = 2 + 5d.So, the three terms in the geometric sequence are 2, 2 + 2d, 2 + 5d. Since these form a geometric sequence, the square of the middle term should equal the product of the first and third terms. So, (2 + 2d)² = 2*(2 + 5d).Let me write that out:(2 + 2d)² = 2*(2 + 5d)Let me compute the left side first. (2 + 2d)² is equal to 2² + 2*2*2d + (2d)², which is 4 + 8d + 4d². So, left side is 4 + 8d + 4d².The right side is 2*(2 + 5d) = 4 + 10d.So, setting left side equal to right side:4 + 8d + 4d² = 4 + 10dHmm, let's subtract 4 from both sides:8d + 4d² = 10dThen, subtract 10d from both sides:8d + 4d² - 10d = 0Which simplifies to:4d² - 2d = 0Factor out a 2d:2d*(2d - 1) = 0So, either 2d = 0 or 2d - 1 = 0. That gives d = 0 or d = 1/2. But the problem states that the common difference is non-zero, so we discard d = 0. Therefore, d = 1/2.Alright, so the common difference is 1/2. Now, we need to find the sum of the first 9 terms of this arithmetic sequence.Recall that the sum of the first n terms of an arithmetic sequence is given by S_n = n/2*(2a₁ + (n - 1)d) or S_n = n*(a₁ + a_n)/2. Either formula works.Let me use the first formula. Here, n = 9, a₁ = 2, d = 1/2.So, S₉ = 9/2*(2*2 + (9 - 1)*(1/2))Compute inside the brackets first: 2*2 is 4, and (9 - 1) is 8, so 8*(1/2) is 4. So, inside the brackets: 4 + 4 = 8.Then, S₉ = 9/2 * 8 = (9*8)/2 = 72/2 = 36.Wait, so that would be 36, which is option C.But let me double-check using the other formula to make sure I didn't make a mistake. The other formula is S₉ = 9*(a₁ + a₉)/2.First, find a₉. Since a₉ = 2 + (9 - 1)*(1/2) = 2 + 8*(1/2) = 2 + 4 = 6.So, S₉ = 9*(2 + 6)/2 = 9*8/2 = 72/2 = 36. Same result. So, that seems consistent.Wait, but let me check again the problem statement. The problem says that the common difference is non-zero, which we considered, so we took d = 1/2. Then, the sum is 36, which is option C. However, the answer choices include 36 as C. So, that seems correct.But let me check again the step where we set up the geometric sequence condition to make sure I didn't make a mistake there.Given that a₁, a₃, a₆ are in geometric progression, so:a₃ / a₁ = a₆ / a₃Which is equivalent to (a₃)² = a₁*a₆Which is the same as (2 + 2d)² = 2*(2 + 5d)Expanding left side: 4 + 8d + 4d². Right side: 4 + 10d.Then, moving everything to one side:4d² + 8d + 4 - 4 - 10d = 0Simplifies to:4d² - 2d = 0Factor out 2d:2d(2d - 1) = 0Solutions d = 0 or d = 1/2. As before.So that's correct.Therefore, the common difference is 1/2, and the sum of the first 9 terms is 36, so the answer is C.But let me check the answer options again. The options are A:26, B:30, C:36, D:40. So yes, 36 is option C.But just to make sure, let me list out the first 9 terms and compute the sum manually.Given a₁ = 2, d = 1/2.So, a₁ = 2a₂ = 2 + 1/2 = 2.5a₃ = 2 + 2*(1/2) = 2 + 1 = 3a₄ = 2 + 3*(1/2) = 2 + 1.5 = 3.5a₅ = 2 + 4*(1/2) = 2 + 2 = 4a₆ = 2 + 5*(1/2) = 2 + 2.5 = 4.5a₇ = 2 + 6*(1/2) = 2 + 3 = 5a₈ = 2 + 7*(1/2) = 2 + 3.5 = 5.5a₉ = 2 + 8*(1/2) = 2 + 4 = 6Now, sum these up:2 + 2.5 + 3 + 3.5 + 4 + 4.5 + 5 + 5.5 + 6Let me add them step by step:Start with 2.2 + 2.5 = 4.54.5 + 3 = 7.57.5 + 3.5 = 1111 + 4 = 1515 + 4.5 = 19.519.5 + 5 = 24.524.5 + 5.5 = 3030 + 6 = 36.Yep, that's 36. So that checks out. Therefore, the answer is C:36.But hold on, the problem gives options, and 36 is one of them. So, seems correct.Wait, but in the problem statement, it says "the sum of the first 9 terms of {a_n} is (　　)". So, answer is 36, option C. So, that's correct.Therefore, I think the correct answer is C:36.Final Answerboxed{C}</think>Given that ({a_n}) is an arithmetic sequence with the first term (a_1 = 2) and a non-zero common difference (d), we need to find the sum of the first 9 terms if (a_1), (a_3), (a_6) form a geometric sequence.First, we express the terms (a_1), (a_3), and (a_6) in terms of (d):- (a_1 = 2)- (a_3 = 2 + 2d)- (a_6 = 2 + 5d)Since these terms form a geometric sequence, we use the property of geometric sequences that the square of the middle term equals the product of the first and third terms:[(2 + 2d)^2 = 2 cdot (2 + 5d)]Expanding and simplifying:[4 + 8d + 4d^2 = 4 + 10d][4d^2 - 2d = 0][2d(2d - 1) = 0]Since (d neq 0), we solve for (d):[2d - 1 = 0 implies d = frac{1}{2}]Now, we find the sum of the first 9 terms of the arithmetic sequence with (a_1 = 2) and (d = frac{1}{2}). The sum (S_n) of the first (n) terms of an arithmetic sequence is given by:[S_n = frac{n}{2} left(2a_1 + (n - 1)d right)]Substituting (n = 9), (a_1 = 2), and (d = frac{1}{2}):[S_9 = frac{9}{2} left(2 cdot 2 + 8 cdot frac{1}{2} right)][S_9 = frac{9}{2} left(4 + 4 right)][S_9 = frac{9}{2} cdot 8 = 36]Thus, the sum of the first 9 terms is (boxed{C}).,<think>Okay, let's see. So the problem says that we have an arithmetic sequence {a_n} with the first term being 2 and a non-zero common difference. Then it says that a_1, a_3, a_6 form a geometric sequence. We need to find the sum of the first 9 terms of {a_n}, and the options are A:26, B:30, C:36, D:40.First, let's recall what an arithmetic sequence is. An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is a_1 and the common difference is d, then the nth term is a_n = a_1 + (n - 1)d. In this case, a_1 is 2, so the nth term would be 2 + (n - 1)d.Now, we need to consider the terms a_1, a_3, a_6 forming a geometric sequence. A geometric sequence is one where each term after the first is found by multiplying the previous term by a constant ratio, called the common ratio. So, if a_1, a_3, a_6 are in geometric sequence, then the ratio between a_3 and a_1 should be the same as the ratio between a_6 and a_3.Mathematically, this means that a_3 / a_1 = a_6 / a_3. So, cross-multiplying, we get (a_3)^2 = a_1 * a_6.Let's compute each term:a_1 is given as 2.a_3, being the third term of the arithmetic sequence, would be a_1 + 2d, which is 2 + 2d.Similarly, a_6 is the sixth term, which is a_1 + 5d, so that's 2 + 5d.So, substituting these into the equation (a_3)^2 = a_1 * a_6, we have:(2 + 2d)^2 = 2 * (2 + 5d)Let's expand the left side:(2 + 2d)^2 = 4 + 8d + 4d^2And the right side is 2*(2 + 5d) = 4 + 10dSo, setting them equal:4 + 8d + 4d^2 = 4 + 10dSubtract 4 from both sides:8d + 4d^2 = 10dSubtract 10d from both sides:4d^2 - 2d = 0Factor out 2d:2d(2d - 1) = 0So, either 2d = 0 or 2d - 1 = 0But the problem states that the common difference is non-zero, so 2d = 0 would lead to d = 0, which is invalid. Therefore, we take the other solution:2d - 1 = 0 => 2d = 1 => d = 1/2So, the common difference d is 1/2.Now, we need to find the sum of the first 9 terms of the arithmetic sequence.Recall that the sum of the first n terms of an arithmetic sequence is given by:S_n = n/2 * [2a_1 + (n - 1)d]Alternatively, S_n = n * (a_1 + a_n)/2Either formula can be used. Let's use the first one.Here, n = 9, a_1 = 2, d = 1/2.So, S_9 = 9/2 * [2*2 + (9 - 1)*(1/2)]Calculate inside the brackets:2*2 = 4(9 - 1) = 8, so 8*(1/2) = 4So, inside the brackets: 4 + 4 = 8Therefore, S_9 = (9/2) * 8 = (9 * 8)/2 = 72/2 = 36Therefore, the sum of the first 9 terms is 36, which is option C.Wait a second, but let me check my calculations again to make sure I didn't make a mistake.First, we found that d = 1/2.So, the terms are:a_1 = 2a_2 = 2 + 1/2 = 2.5a_3 = 3.0a_4 = 3.5a_5 = 4.0a_6 = 4.5Wait, let's check if a_1, a_3, a_6 form a geometric sequence.a_1 = 2a_3 = 2 + 2d = 2 + 2*(1/2) = 2 + 1 = 3a_6 = 2 + 5d = 2 + 5*(1/2) = 2 + 2.5 = 4.5So, the three terms are 2, 3, 4.5. Let's check if they form a geometric sequence.The ratio between a_3 and a_1 is 3/2 = 1.5The ratio between a_6 and a_3 is 4.5/3 = 1.5So yes, the common ratio is 1.5, so that's correct.Now, sum of the first 9 terms.Another way to compute the sum is using the formula S_n = n/2*(first term + last term). Let's compute the ninth term.a_9 = 2 + (9 - 1)*d = 2 + 8*(1/2) = 2 + 4 = 6So, the first term is 2, the ninth term is 6. Therefore, the sum is 9/2*(2 + 6) = (9/2)*8 = 36. So that's the same result. Therefore, the answer is 36, which is option C. So that should be the correct answer.But wait, let me check again if there could be another solution. When we solved the equation, we had 4d^2 - 2d = 0, which factors to 2d(2d - 1) = 0. The solutions are d = 0 or d = 1/2. Since d is non-zero, d = 1/2 is the only solution. So, that seems correct.Alternatively, perhaps the problem is designed to have a trick, but I don't see it. Let's check the options again. The sum is 36, which is option C. So, I think that's the correct answer.Wait, just to be thorough, let's compute the sum manually by adding all nine terms.a_1 = 2a_2 = 2 + 1/2 = 2.5a_3 = 3a_4 = 3.5a_5 = 4a_6 = 4.5a_7 = 5a_8 = 5.5a_9 = 6Adding these up:2 + 2.5 = 4.54.5 + 3 = 7.57.5 + 3.5 = 1111 + 4 = 1515 + 4.5 = 19.519.5 + 5 = 24.524.5 + 5.5 = 3030 + 6 = 36Yes, adding them all up gives 36. So that's correct.Therefore, the answer is C: 36.Final Answerboxed{C}</think>Given that ({a_n}) is an arithmetic sequence with the first term being 2 and a non-zero common difference (d), we need to find the sum of the first 9 terms if (a_1), (a_3), (a_6) form a geometric sequence.First, we determine the terms of the arithmetic sequence:- (a_1 = 2)- (a_3 = 2 + 2d)- (a_6 = 2 + 5d)Since (a_1), (a_3), (a_6) form a geometric sequence, we have:[left( a_3 right)^2 = a_1 cdot a_6]Substituting the terms, we get:[(2 + 2d)^2 = 2 cdot (2 + 5d)]Expanding and simplifying:[4 + 8d + 4d^2 = 4 + 10d][4d^2 - 2d = 0]Factoring out (2d):[2d(2d - 1) = 0]Since (d neq 0), we have (2d - 1 = 0 implies d = frac{1}{2}).Next, we calculate the sum of the first 9 terms of the arithmetic sequence. The sum (S_n) of the first (n) terms of an arithmetic sequence is given by:[S_n = frac{n}{2} left[ 2a_1 + (n - 1)d right]]Substituting (n = 9), (a_1 = 2), and (d = frac{1}{2}):[S_9 = frac{9}{2} left[ 2 cdot 2 + 8 cdot frac{1}{2} right]][S_9 = frac{9}{2} left[ 4 + 4 right]][S_9 = frac{9}{2} cdot 8 = frac{72}{2} = 36]Thus, the sum of the first 9 terms is (boxed{C}).]

🔑:Let the common difference be d, Since {a_n} is an arithmetic sequence with the first term being 2 and a non-zero common difference, and a_1, a_3, a_6 form a geometric sequence, Therefore, (2+2d)^2 = 2(2+5d), and d neq 0, Solving this, we get d = frac{1}{2}, Therefore, the sum of the first 9 terms of {a_n}, S_9 = 9 times 2 + frac{9 times 8}{2} times frac{1}{2} = 36. Hence, the correct choice is: boxed{C}. Let the common difference be d, from the given information (2+2d)^2 = 2(2+5d), and d neq 0, solving this, we get d = frac{1}{2}, from which we can calculate the sum of the first 9 terms of {a_n}. This problem tests the method of finding the sum of the first 9 terms of an arithmetic sequence, which is a basic question. When solving the problem, it is important to carefully read the question and properly apply the properties of arithmetic and geometric sequences.

🧠:[<think>Okay, so I need to calculate this big expression: 20130 plus 2 times (480 times 4 times 14 plus 30 times 44 times 16) minus (5 times 808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080808080800000,<think>Okay, so I need to calculate this big expression: 20130 plus 2 times (480 times 4 times 14 plus 30 times 44 times 16) minus (5 times 80 times 43 plus 19 times 400 times 3) times 2. Hmm, that's a lot. Let me break it down step by step. First, I should remember the order of operations: parentheses first, then multiplication and division, then addition and subtraction. So, looking at the expression, there are several parts inside parentheses that I need to calculate first. Let me write down the expression again to visualize it better:20130 + 2*(480*4*14 + 30*44*16) - (5*80*43 + 19*400*3)*2Alright, so starting with the first part inside the parentheses: 480*4*14. Let me calculate that. 480 multiplied by 4. Let me do 480*4 first. 480*4 is 1920. Then multiply that by 14. Hmm, 1920*14. Let me break that down. 1920*10 is 19200, and 1920*4 is 7680. So adding those together, 19200 + 7680 = 26880. So the first part inside the first parentheses is 26880.Next part inside the first parentheses is 30*44*16. Let me compute that. First, 30*44. 30*40 is 1200, and 30*4 is 120, so 1200+120=1320. Then multiply that by 16. 1320*16. Let's break it down: 1000*16=16000, 300*16=4800, 20*16=320. So 16000+4800=20800, plus 320 is 21120. Wait, wait, hold on. Wait, 1320 is 1000 + 300 + 20. So 1000*16=16000, 300*16=4800, 20*16=320. Adding those: 16000 + 4800 = 20800, plus 320 is 21120. Yes, that's correct. So 30*44*16 is 21120.Now, adding those two results together: 26880 + 21120. Let's see, 26880 + 21120. 26k + 21k is 47k, 880 + 120 is 1000. So total is 47k + 1k = 48,000. Wait, really? 26880 + 21120 = 48,000? Let me check: 26,880 + 21,120. 26,880 + 20,000 = 46,880, then +1,120 = 48,000. Yes, that's correct. So the first big parentheses is 48,000. Then multiply that by 2. So 48,000 * 2 = 96,000.So the first part of the expression is 20130 + 96,000. Let's compute that. 20,130 + 96,000. 20k + 96k = 116k, and then 130. So 116,130. Got that.Now, moving to the second part: (5*80*43 + 19*400*3)*2. Again, let's compute inside the parentheses first. First term: 5*80*43. Let's compute 5*80 first. 5*80=400. Then 400*43. Hmm, 400*40=16,000 and 400*3=1,200. So 16,000 + 1,200 = 17,200. So the first term is 17,200.Second term: 19*400*3. Let's compute 19*400 first. 19*400. 10*400=4,000, 9*400=3,600. So 4,000 + 3,600 = 7,600. Then multiply by 3: 7,600*3. 7,000*3=21,000, 600*3=1,800. So 21,000 + 1,800 = 22,800. So the second term is 22,800.Adding those two results: 17,200 + 22,800. 17k + 22k = 39k, 200 + 800 = 1,000. So total is 40,000. Wait, 17,200 + 22,800. Let me check: 17,200 + 22,000 = 39,200, then +800 = 40,000. Yes, that's correct. So the inside of the parentheses is 40,000. Then multiply that by 2: 40,000*2 = 80,000.Now, putting it all back into the original expression: 20130 + 96,000 - 80,000. So 20130 + (96,000 - 80,000). Let's compute 96,000 - 80,000 first, which is 16,000. Then add that to 20,130. So 20,130 + 16,000. That's 36,130. Wait, hold on. Wait, no. Wait, the original expression is 20130 + 96,000 - 80,000. So addition and subtraction from left to right. So first, 20,130 + 96,000 = 116,130. Then subtract 80,000: 116,130 - 80,000 = 36,130. Hmm, that's different from what I thought earlier. Wait, no, 20130 plus 96000 is 116,130. Then subtract 80,000. 116,130 minus 80,000 is 36,130. So the final result is 36,130.Wait, but let me double-check all the steps to make sure I didn't make a mistake anywhere. So first, 480*4*14. 480*4=1920, 1920*14=26,880. Correct. Then 30*44=1,320, 1,320*16=21,120. Then 26,880 +21,120=48,000. Multiply by 2: 96,000. Then 20,130 +96,000=116,130. Then the second part: 5*80=400, 400*43=17,200. Then 19*400=7,600, 7,600*3=22,800. Then 17,200 +22,800=40,000. Multiply by 2: 80,000. Then 116,130 -80,000=36,130. Hmm. Seems correct.Wait, but let me check each multiplication again just to be sure. Let's start with 480*4*14. 480*4: 4*400=1600, 4*80=320, so 1600+320=1920. Then 1920*14: 1920*10=19,200; 1920*4=7,680. Adding them: 19,200 +7,680=26,880. Correct. Then 30*44*16: 30*44: 30*40=1,200, 30*4=120, so 1,200+120=1,320. Then 1,320*16: 1,320*10=13,200; 1,320*6=7,920. So 13,200 +7,920=21,120. Correct. Then 26,880 +21,120: 26,880 +20,000=46,880; 46,880 +1,120=48,000. Correct. Then 48,000*2=96,000. Then 20,130 +96,000=116,130. Correct.Now the second part: 5*80*43: 5*80=400. Then 400*43: 400*40=16,000; 400*3=1,200. 16,000 +1,200=17,200. Correct. Then 19*400*3: 19*400. 10*400=4,000; 9*400=3,600; total 7,600. Then 7,600*3=22,800. Correct. Then 17,200 +22,800=40,000. Correct. 40,000*2=80,000. Then 116,130 -80,000=36,130. Hmm. So seems correct.But let me check once more. Maybe I made a mistake in the order? The original expression is:20130 + 2*(...) - (...)*2So according to order of operations, multiplication comes before addition and subtraction. But since these terms are already computed, it's just left to right for the addition and subtraction. So 20130 + [2*(...)] - [(...)*2]. So yes, first compute both multiplied terms, then add and subtract. So 20130 + 96,000 - 80,000 = 36,130. That's the result.But let me verify with another approach. Maybe compute the entire expression step by step without breaking it into parts. Let's see.Original expression:20130 + 2*(480*4*14 + 30*44*16) - (5*80*43 + 19*400*3)*2First, compute all the multiplications inside the first parentheses:480*4*14: 480*4=1920; 1920*14=26,88030*44*16: 30*44=1,320; 1,320*16=21,120Add them: 26,880 +21,120=48,000Multiply by 2: 48,000*2=96,000Then compute the second parentheses:5*80*43: 5*80=400; 400*43=17,20019*400*3: 19*400=7,600; 7,600*3=22,800Add them:17,200 +22,800=40,000Multiply by 2:40,000*2=80,000Now the expression is:20130 +96,000 -80,000Which is 20130 +16,000=36,130Yes, same result. So I think that's correct.Alternatively, maybe I can compute the entire expression in another way. Let me check with another grouping.But probably not necessary. All steps check out. So the answer is 36,130.Final Answerboxed{36130}</think>To calculate the given expression:20130 + 2 cdot (480 cdot 4 cdot 14 + 30 cdot 44 cdot 16) - (5 cdot 80 cdot 43 + 19 cdot 400 cdot 3) cdot 2we break it down step by step:1. Calculate the terms inside the first parentheses: - (480 cdot 4 cdot 14): [ 480 cdot 4 = 1920 1920 cdot 14 = 26880 ] - (30 cdot 44 cdot 16): [ 30 cdot 44 = 1320 1320 cdot 16 = 21120 ] - Sum the results: [ 26880 + 21120 = 48000 ] - Multiply by 2: [ 48000 cdot 2 = 96000 ]2. Calculate the terms inside the second parentheses: - (5 cdot 80 cdot 43): [ 5 cdot 80 = 400 400 cdot 43 = 17200 ] - (19 cdot 400 cdot 3): [ 19 cdot 400 = 7600 7600 cdot 3 = 22800 ] - Sum the results: [ 17200 + 22800 = 40000 ] - Multiply by 2: [ 40000 cdot 2 = 80000 ]3. Combine all parts of the expression: - Start with 20130, add 96000, then subtract 80000: [ 20130 + 96000 = 116130 116130 - 80000 = 36130 ]Thus, the final answer is:[boxed{36130}]]

🔑:Solution. By direct calculation, we getbegin{aligned}20130+2 cdot(480 cdot 4 cdot 14 & +30 cdot 44 cdot 16)-(5 cdot 80 cdot 43+19 cdot 400 cdot 3) cdot 2= & =20130+2 cdot(480 cdot 56+44 cdot 480)-(400 cdot 43+400 cdot 57) cdot 2 & =20130+2 cdot 480 cdot 100-2 cdot 400 cdot 100 & =20130+96000-80000 & =20130+16000=36130end{aligned}

🧠:[<think>Okay, so I need to find the radius of a circle where a diameter and a chord intersect at a point inside the circle. The chord is split into parts of length 3 and 5 by this intersection, and one part of the diameter is length 1. Hmm, let's try to visualize this first. First, let me recall some geometry concepts. When two chords intersect in a circle, there's a theorem that says the products of the segments of each chord are equal. Wait, is that right? Let me check. If two chords intersect at a point, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. So, if we have chord AB intersecting chord CD at point E inside the circle, then AE * EB = CE * ED. In this problem, one of the chords is a diameter. Let me denote the diameter as AB and the chord as CD, intersecting at point E. The chord CD is divided into parts CE = 3 and ED = 5. The diameter AB is divided into two parts by E, but we know only one part is length 1. Wait, the problem says "one part of the diameter is length 1". So, either AE = 1 or EB = 1. Since AB is a diameter, the entire length is 2r, where r is the radius. So, if one part is 1, the other part would be 2r - 1. But according to the intersecting chords theorem, we have AE * EB = CE * ED. So, if one segment of the diameter is 1, then the other segment would be (CE * ED)/1 = 3*5 = 15. So, 1 * 15 = 15, which would mean that the diameter is 1 + 15 = 16, so the radius is 8? Wait, that seems straightforward, but let me make sure I'm not missing anything here.Wait, but hold on. The problem states that the diameter and chord intersect at a point inside the circle. The chord is split into 3 and 5. But the diameter is split into 1 and something else. So, according to the theorem, the product of the two parts of the diameter must equal the product of the two parts of the chord. So, AE * EB = CE * ED. If one part of the diameter is 1, then AE = 1 and EB = x. So, 1 * x = 3 * 5 = 15. Therefore, x = 15. Then the total diameter is 1 + 15 = 16, so radius is 8. Is that the answer? It seems too straightforward, but maybe that's right. Wait, but let me think again. Is there a different way to interpret the problem? The problem says "one part of the diameter is length 1". So, maybe the diameter is split into 1 and another part, but which part is it? The part closer to the center or further away? Wait, in the case of a diameter, the center is the midpoint. If the intersection point is inside the circle, then depending on where E is, the segments of the diameter could be different. If the center is O, then the diameter AB is split by E into two parts, AE and EB. If E is not the center, then one of AE or EB is longer than the other. If one part is 1, then the other part is 2r - 1. But according to the theorem, AE * EB = CE * ED. So, if AE = 1, then EB = 2r - 1, so 1*(2r - 1) = 3*5 =15. Then 2r -1 =15, so 2r =16, so r=8. Alternatively, if EB =1, then AE =2r -1, so (2r -1)*1=15, which would again lead to 2r -1=15, so 2r=16, r=8. So either way, the radius is 8. Wait, that seems to make sense. So regardless of which segment of the diameter is 1, the result is the same? Because if AE is 1, then EB is 2r -1, product is 1*(2r -1) =15. If EB is1, then AE is 2r -1, product is same. So either way, same equation leading to r=8. So that's the answer. But let me check if I'm applying the theorem correctly. The theorem is when two chords intersect inside the circle, the products of their respective segments are equal. So, in this case, the two chords are the diameter AB and the chord CD. So AE * EB = CE * ED. Which is exactly what I used. So AE * EB = 3 *5=15. Therefore, regardless of which part of the diameter is 1, we have the product 1*(2r -1)=15. Solving gives r=8. But just to make sure, let's visualize the problem again. Imagine a circle with center O. Let's say AB is a diameter, so O is the midpoint. Then there's a chord CD that intersects AB at point E. CE is 3, ED is5. And one of the segments of AB (either AE or EB) is 1. If E is closer to one end, say A, then AE=1, EB=2r -1. Then by the theorem, 1*(2r -1)=15. So 2r=16, so r=8. Alternatively, if E is closer to B, EB=1, then AE=2r -1, which again gives same equation. So in both cases, radius is 8. But perhaps there's another way to approach this problem, maybe coordinate geometry? Let me try that as a check. Let's set up a coordinate system. Let the center of the circle be at (0,0). Let the diameter AB be along the x-axis, so A is (-r,0) and B is (r,0). Let the chord CD intersect AB at point E. Let’s say E is at (h,0). Since one part of the diameter is 1, the distance from E to either A or B is 1. Suppose E is 1 unit away from A, then AE=1, so the coordinate of E is (-r +1,0). Wait, but the length from A to E is 1, so since A is (-r,0), E would be at (-r +1,0). Then EB, the distance from E to B, is r - (-r +1) = 2r -1. Alternatively, if E is 1 unit away from B, then EB=1, so coordinate of E is (r -1,0), and AE would be (r -1) - (-r) = 2r -1. Either way, we have the same as before. Now, the chord CD passes through E, which is either (-r +1,0) or (r -1,0). Let's suppose we take E as (h,0), with AE=1, so h = -r +1. The chord CD has segments CE=3 and ED=5, so total length CD=8. Let’s parametrize chord CD. Let’s suppose CD is a vertical line passing through E, but since the circle is symmetric, we can assume CD is perpendicular to AB? Wait, but CD could be at an angle. Wait, no, unless specified, the chord can be at any angle. So maybe we need to find the coordinates of points C and D such that CE=3 and ED=5. Alternatively, since we know that CD is a chord of the circle, and E is a point inside the circle where CD intersects the diameter AB. So, coordinates of C and D must lie on the circle. Let me set up coordinates. Let’s take E at (h,0), and CD as a vertical line passing through E. Wait, but maybe it's not vertical. Hmm, perhaps better to parametrize CD. Let’s assume that CD is not perpendicular to AB. Let’s denote point E as (h,0). Let’s denote the coordinates of C as (h + x, y) and D as (h - x, -y), such that E is the midpoint between C and D? Wait, no, E divides CD into CE=3 and ED=5, so the midpoint is not E. Instead, the coordinates of C and D can be found by considering that the distance from E to C is 3 and from E to D is 5. Let me denote E as (h,0). Then, let’s set up coordinates such that C is (h + 3*cosθ, 0 + 3*sinθ) and D is (h -5*cosθ, 0 -5*sinθ), since moving from E in the opposite direction for D. Since points C and D lie on the circle centered at (0,0) with radius r. So, coordinates of C: (h + 3*cosθ, 3*sinθ) must satisfy the circle equation: (h + 3*cosθ)^2 + (3*sinθ)^2 = r^2. Similarly, coordinates of D: (h -5*cosθ, -5*sinθ) must satisfy (h -5*cosθ)^2 + (-5*sinθ)^2 = r^2. So, we have two equations:1) (h + 3*cosθ)^2 + 9*sin²θ = r²2) (h -5*cosθ)^2 + 25*sin²θ = r²Since both equal to r², set them equal:(h + 3*cosθ)^2 + 9*sin²θ = (h -5*cosθ)^2 + 25*sin²θLet’s expand both sides:Left side: h² + 6h*cosθ + 9*cos²θ + 9*sin²θRight side: h² -10h*cosθ +25*cos²θ +25*sin²θSubtract left side from right side:(h² -10h*cosθ +25*cos²θ +25*sin²θ) - (h² +6h*cosθ +9*cos²θ +9*sin²θ) =0Simplify:-16h*cosθ +16*cos²θ +16*sin²θ=0Factor out 16:16*(-h*cosθ + cos²θ + sin²θ)=0Since 16≠0, we have:-h*cosθ + (cos²θ + sin²θ)=0But cos²θ + sin²θ=1, so:-h*cosθ +1=0Thus:h*cosθ=1So h*cosθ=1. That's one equation. Now, we also know that AE=1 or EB=1. Let's suppose AE=1. Then, since A is (-r,0), E is (h,0). The distance AE is |h - (-r)| = |h + r| =1. So h + r=1, since E is between A and B (inside the circle), so h > -r and h < r. Therefore, h + r=1 implies h=1 - r.Similarly, if EB=1, then the distance from E to B is |r - h|=1. Since h < r, then r - h=1, so h=r -1.But in either case, we can proceed. Let's consider first AE=1, so h=1 - r.From the previous equation, h*cosθ=1. So (1 - r)*cosθ=1. Therefore, cosθ=1/(1 - r). But cosθ must be between -1 and 1. So 1/(1 - r) must be in [-1,1]. Let's check:If h=1 - r, and since E is inside the circle, |h| < r. So |1 - r| < r. Since r is positive, this implies:Case 1: 1 - r < r and 1 - r > -r.First inequality: 1 - r < r => 1 < 2r => r > 1/2.Second inequality: 1 - r > -r => 1 > 0, which is always true. So for h=1 - r to be inside the circle, r > 1/2.Similarly, if EB=1, h=r -1. Then |h| < r implies |r -1| < r. So:Case 2: If r -1 >=0 (i.e., r >=1), then r -1 < r => -1 <0, which is true.If r -1 <0 (i.e., r <1), then |r -1|=1 - r < r => 1 - r < r => 1 < 2r => r > 1/2.So in either case, similar to before, r >1/2. But since the chord segments are 3 and 5, the chord has length 8, so the radius must be at least 4 (since the maximum distance between two points is the diameter, which is 2r). So r must be >=4. So we can proceed with that in mind.Back to the case where AE=1, h=1 - r, and cosθ=1/(1 - r). Since r >=4, 1 - r <= -3, so 1/(1 - r) is negative and between -1/3 and 0. So cosθ is between -1/3 and 0. Now, let's use the coordinates of point C or D in the circle equation. Let's take point C: (h +3*cosθ, 3*sinθ). Plugging into the circle equation:(h +3*cosθ)^2 + (3*sinθ)^2 = r^2Expand:h² +6h*cosθ +9*cos²θ +9*sin²θ = r²We can write this as:h² +6h*cosθ +9(cos²θ + sin²θ) = r²Again, cos²θ + sin²θ=1, so:h² +6h*cosθ +9 = r²But we know from earlier that h*cosθ=1. So:h² +6*1 +9 = r²So h² +15 = r²But h=1 - r (since we considered AE=1), so substitute:(1 - r)^2 +15 = r²Expand:1 -2r + r² +15 = r²Simplify:16 -2r =0Therefore:-2r = -16 => r=8So that's consistent with the earlier result. Alternatively, if we consider EB=1, then h=r -1. From h*cosθ=1, we have (r -1)*cosθ=1, so cosθ=1/(r -1). Since r >=4, r -1 >=3, so cosθ=1/(r -1) is positive and <=1/3. So okay, that's valid.Using the same equation for point C:h² +6h*cosθ +9 = r²But h=r -1 and h*cosθ=1, so:(r -1)^2 +6*1 +9 = r²Expand:r² -2r +1 +6 +9 =r²Simplify:r² -2r +16 =r²Subtract r²:-2r +16=0 => -2r= -16 => r=8Same result. Therefore, regardless of whether AE=1 or EB=1, the radius is 8. Therefore, the radius of the circle is 8. I think this confirms the answer. First, using the intersecting chords theorem gives us the answer directly, and then using coordinate geometry with parametrization also leads to the same result. So I feel confident that the radius is 8.Final AnswerThe radius of the circle is boxed{8}.</think>Given a circle where a diameter and a chord intersect at a point inside the circle, we need to find the radius of the circle. The chord is divided into segments of lengths 3 and 5, and one part of the diameter is length 1.Using the intersecting chords theorem, which states that the product of the segments of one chord is equal to the product of the segments of the other chord, we have:[AE times EB = CE times ED]Given (CE = 3), (ED = 5), and one segment of the diameter is 1, let the radius be (r). The diameter (AB) is (2r). If one segment of the diameter is 1, the other segment must be (2r - 1). Applying the theorem:[1 times (2r - 1) = 3 times 5]Solving this equation:[2r - 1 = 15][2r = 16][r = 8]To verify, we use coordinate geometry. Setting the center of the circle at ((0,0)) and considering the diameter along the x-axis, we analyze the coordinates of the intersection point (E). Whether (E) is 1 unit from (A) or (B), the product of the segments of the diameter and the chord leads to the same result. Solving the equations for the circle and the coordinates of the chord's endpoints confirms the radius is indeed 8.Thus, the radius of the circle is (boxed{8}).,<think>Okay, so I need to find the radius of a circle where a diameter and a chord intersect at a point inside the circle. The chord is divided into parts of lengths 3 and 5, and one part of the diameter is length 1. Hmm, let me visualize this.First, let me recall some geometry theorems that might help here. When two chords intersect in a circle, there's a theorem that says the products of the segments of each chord are equal. Wait, is that applicable here? Let me check.The problem says a diameter and a chord intersect. The diameter is a specific chord that passes through the center, right? So when they intersect inside the circle, the theorem about intersecting chords should apply here. The theorem states that if two chords intersect at a point, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.In this case, one chord is the diameter, which is split into two parts at the intersection point. Let's denote the lengths of the two parts of the diameter as ( x ) and ( y ). The chord is split into parts of 3 and 5. According to the theorem, we should have ( x times y = 3 times 5 = 15 ). But we are told that one part of the diameter is length 1. Wait, which part? Is it the part from the intersection point to the circumference, or from the center to the intersection point?Hmm, this might be a point of confusion. Let me think. The diameter is split into two parts by the intersection point. Since the diameter passes through the center, one of the segments will be from the center to the intersection point, and the other will be from the intersection point to the circumference. But the problem states "one part of the diameter is length 1". It doesn't specify which part. So I need to consider both possibilities.Let me denote the distance from the center to the intersection point as ( a ), and the distance from the intersection point to the circumference as ( b ). Then the diameter's total length is ( 2r ), where ( r ) is the radius. Therefore, ( a + b = 2r ). But according to the problem, one part of the diameter is length 1. So either ( a = 1 ) or ( b = 1 ).But wait, the chord intersects the diameter. The chord is split into segments of 3 and 5. The theorem tells us that ( a times b = 3 times 5 = 15 ). Wait, is that correct? No, hold on. The theorem actually states that if two chords intersect, then the products of the segments of each chord are equal. So in this case, the two chords are the diameter and the chord in question. Therefore, the products would be ( (segment1 of diameter) times (segment2 of diameter) = (segment1 of chord) times (segment2 of chord) ).So, in symbols: if the diameter is divided into two segments of lengths ( m ) and ( n ), then ( m times n = 3 times 5 = 15 ). But since the diameter is passing through the center, one of the segments ( m ) or ( n ) must be from the intersection point to the center, and the other from the intersection point to the circumference. Wait, but the entire diameter is ( 2r ), so if one segment is length 1, then the other is ( 2r - 1 ). Therefore, ( 1 times (2r - 1) = 15 ). Solving for ( r ), we get ( 2r - 1 = 15 ), so ( 2r = 16 ), so ( r = 8 ).But hold on, is this right? Let me confirm. If one part of the diameter is 1, then the other part must be ( 2r - 1 ). Then, by the intersecting chords theorem, ( 1 times (2r - 1) = 3 times 5 ). Then, ( 2r - 1 = 15 ), so ( 2r = 16 ), so ( r = 8 ). That seems straightforward.But wait, is there a different interpretation? The problem says "one part of the diameter is length 1". If "part" refers to one of the two segments created by the intersection, then yes, either the segment from the intersection to the circumference is 1, or from the center to the intersection is 1. Wait, but if the diameter is split into two parts, one of which is 1, then depending on which part it is, the calculation changes.Wait, hold on. Let me clarify. Let's denote the intersection point as P. Then, if the diameter is split into two segments at P: one segment from P to one end of the diameter (length 1), and the other from P to the other end (length something else). But since the diameter must pass through the center, the distance from P to the center must be part of that. So perhaps the segment of length 1 is from P to the center, or from P to the circumference.If the length from P to the center is 1, then the length from P to the circumference would be ( 2r - 1 ). But if the length from P to the circumference is 1, then the length from P to the center is ( 2r - 1 ). Wait, but in both cases, the product of the two segments would be ( 1 times (2r - 1) = 15 ), so either way, the equation is the same. Therefore, solving for ( r ), we get ( r = 8 ).But is this possible? Let's check if the radius is 8, then the diameter is 16. If one part of the diameter is 1, the other part is 15. Then, according to the intersecting chords theorem, 1*15=3*5=15. So it holds.But wait, let me verify with coordinates. Let me set up a coordinate system where the center of the circle is at (0,0), and the diameter is along the x-axis. Let the intersection point P be at (h,0). Then the two ends of the diameter are (-8,0) and (8,0). The chord intersects the diameter at point P, which is distance 1 from somewhere.Wait, if the segment from P to the center is 1, then h = 1, so P is at (1,0). Then, the other segment of the diameter from P to the circumference is from (1,0) to (8,0), which is length 7. But 1*7 = 7, which is not 15. That contradicts the theorem.Wait, this is a problem. So my previous reasoning must have a mistake. Wait, so if the center is at (0,0), and if the intersection point is at (h,0), then the two segments of the diameter are |h - (-8)| = h + 8 and |8 - h| = 8 - h. Wait, no. If the diameter is from (-8,0) to (8,0), then the intersection point P is at (h,0). Then the two parts of the diameter would be from (-8,0) to (h,0), which is length h + 8, and from (h,0) to (8,0), which is 8 - h. So the product is (h + 8)(8 - h) = 64 - h².According to the theorem, this product should be equal to the product of the chord segments, which is 3*5=15. Therefore, 64 - h² = 15, so h² = 64 - 15 = 49, so h = 7 or h = -7. But since h is a point along the diameter from -8 to 8, h is between -8 and 8. So h = 7 or h = -7.But the problem says one part of the diameter is length 1. If h = 7, then the two parts of the diameter are h + 8 = 15 and 8 - h = 1. So the part from (7,0) to (8,0) is length 1. Similarly, if h = -7, the parts are h + 8 = 1 and 8 - h = 15. So either way, one part is 1 and the other is 15. Therefore, the radius is 8, as before. But earlier when I tried to visualize with coordinates, I thought that h = 1, but that was a misunderstanding.Wait, so the intersection point is at h = 7 or h = -7. So the distance from the center (0,0) to P (7,0) is 7, which is not 1. But the problem states that one part of the diameter is length 1. So if h = 7, then the segment from h to 8 is 1, and from -8 to h is 15. So the part from h to 8 is 1. Therefore, "one part of the diameter is length 1" refers to one of the two segments created by the intersection point. So that part is 1, and the other part is 15. Then, the radius is 8. Wait, but the distance from the center to the intersection point is 7, which is not mentioned in the problem. The problem only mentions that one part of the diameter is 1. So maybe the answer is 8. But let me check if this satisfies the theorem.If the radius is 8, then the diameter is 16. The chord is split into 3 and 5. The intersection divides the diameter into 1 and 15. Then, 1*15 = 3*5 = 15. So that works. Therefore, the radius is 8.But let me just double-check with coordinates. Suppose the circle has center at (0,0) and radius 8. The diameter is along the x-axis. The chord intersects the diameter at (7,0). The chord has segments of length 3 and 5. So the chord is a line passing through (7,0) and intersecting the circle at two points, such that the distances from (7,0) to each point are 3 and 5. Let's find the equation of the chord.Let the chord make an angle θ with the x-axis. Then, the points on the chord can be parametrized as (7 + t*cosθ, 0 + t*sinθ). The points where the chord intersects the circle must satisfy x² + y² = 64. Therefore:(7 + t*cosθ)² + (t*sinθ)² = 64Expanding:49 + 14t*cosθ + t²*cos²θ + t²*sin²θ = 64Simplify:49 + 14t*cosθ + t²*(cos²θ + sin²θ) = 64Since cos²θ + sin²θ = 1:49 + 14t*cosθ + t² = 64So t² + 14t*cosθ + 49 - 64 = 0t² + 14t*cosθ - 15 = 0This quadratic equation in t should have solutions t = 3 and t = -5 (since the chord segments are length 3 and 5 from the intersection point at (7,0)). Wait, but in the parametrization, t is the distance from (7,0) along the chord. So t = 3 and t = -5. Therefore, the quadratic equation would have roots at t = 3 and t = -5. Therefore, the product of the roots is (3)*(-5) = -15. But according to the quadratic equation, the product of the roots is c/a = -15. Which matches. The sum of the roots is 3 + (-5) = -2, which should be equal to -b/a = -14cosθ. Therefore, -14cosθ = -2, so cosθ = 2/14 = 1/7. Therefore, θ = arccos(1/7). So the slope of the chord is tanθ = sqrt(1 - (1/7)^2)/(1/7) = sqrt(48/49)/(1/7) = (4*sqrt(3)/7)/(1/7) = 4*sqrt(3). Therefore, the chord has a slope of 4√3. Therefore, the equation of the chord is y = 4√3(x - 7).Now, let's find the points where this chord intersects the circle x² + y² = 64.Substitute y = 4√3(x - 7) into the circle equation:x² + [4√3(x - 7)]² = 64Compute:x² + 16*3*(x - 7)^2 = 64x² + 48(x² - 14x + 49) = 64x² + 48x² - 672x + 2352 = 6449x² - 672x + 2352 - 64 = 049x² - 672x + 2288 = 0Divide all terms by 49:x² - (672/49)x + 2288/49 = 0Simplify:x² - 13.7142857x + 46.6938776 ≈ 0But this seems messy. Let me use exact values.We had the quadratic equation in t: t² + 14t*(1/7) - 15 = 0, which simplifies to t² + 2t - 15 = 0. Solving this, t = [-2 ± sqrt(4 + 60)]/2 = [-2 ± sqrt(64)]/2 = [-2 ± 8]/2. So t = (6)/2 = 3 or t = (-10)/2 = -5. So the points are at t = 3 and t = -5. Therefore, substituting back into the parametric equations:For t = 3: (7 + 3*(1/7), 0 + 3*sqrt(1 - (1/7)^2)) = (7 + 3/7, 3*sqrt(48/49)) = (52/7, (3*4*sqrt(3))/7) = (52/7, 12√3/7)For t = -5: (7 - 5*(1/7), 0 -5*sqrt(48/49)) = (7 - 5/7, -5*(4√3)/7) = (44/7, -20√3/7)Now, let's compute the distance between these two points to confirm it's 8 (the diameter), but wait, the chord length should be 3 + 5 = 8. Let's check the distance between (52/7, 12√3/7) and (44/7, -20√3/7):Difference in x: (52/7 - 44/7) = 8/7Difference in y: (12√3/7 - (-20√3/7)) = 32√3/7Distance squared: (8/7)^2 + (32√3/7)^2 = 64/49 + (1024*3)/49 = 64/49 + 3072/49 = 3136/49 = 64Therefore, distance is sqrt(64) = 8. Which matches the chord length of 3 + 5 = 8. So everything checks out.Therefore, the radius is indeed 8. So the answer is boxed{8}Final Answerboxed{8}</think>Given a circle where a diameter and a chord intersect at a point inside the circle, we need to find the radius of the circle. The chord is divided into segments of lengths 3 and 5, and one part of the diameter is length 1.Using the intersecting chords theorem, which states that the products of the segments of each chord are equal, we denote the parts of the diameter as ( x ) and ( y ). The theorem gives us the equation:[x times y = 3 times 5 = 15]We know that one part of the diameter is length 1. Let ( x = 1 ). Then the other part ( y ) must be ( 2r - 1 ) (since the total length of the diameter is ( 2r )). Substituting these into the equation:[1 times (2r - 1) = 15]Solving for ( r ):[2r - 1 = 15 2r = 16 r = 8]To verify, we consider the coordinates and equations of the circle and chord. The circle with radius 8 has a diameter of 16. The intersecting point divides the diameter into segments of 1 and 15, and the chord into segments of 3 and 5. Using the coordinates and parametric equations confirms the calculations, showing that the products of the segments are equal and the distances satisfy the given conditions.Thus, the radius of the circle is (boxed{8}).]

🔑:Let the chord be AB, the diameter be CD and the point of intersection X. Then XA.XB = XC.XD, so the other part of the diameter has length 15. So the diameter has length 16 and the radius is 8. 3rd CaMO 1971 © John Scholes [email protected] 10 June 2002

🧠:[<think>Okay, let's try to figure out this problem. So, we have a regular triangle, a square, and a regular hexagon, all with the same perimeter. Their areas are S₃, S₄, S₆ respectively. We need to compare these areas and choose the correct option from A to E. First, I remember that for regular polygons with the same perimeter, the one with more sides usually has a larger area. So, a hexagon should have a larger area than a square, which in turn should be larger than a triangle. But wait, the options don't all say that. Let me check again. The options are (A) S₃ > S₄ > S₆; (B) S₆ > S₄ > S₃; (C) S₆ > S₃ > S₄; (D) S₃ > S₆ > S₄; (E) S₄ > S₆ > S₃. Hmm, my initial thought was that hexagon (6 sides) would be largest, then square (4 sides), then triangle (3 sides), so S₆ > S₄ > S₃, which is option B. But maybe I'm oversimplifying?Wait, maybe it's not just the number of sides, but the actual calculation. Let me verify by calculating each area step by step. Since the perimeters are the same, let's assume a common perimeter for all three shapes. Let's choose a perimeter that's a multiple of 3, 4, and 6 to make calculations easier. Maybe 12 units? That way, the triangle would have sides of 12/3 = 4 units each, the square 12/4 = 3 units each, and the hexagon 12/6 = 2 units each. Starting with the regular triangle (equilateral triangle). The formula for the area of an equilateral triangle is (√3/4) * side². So plugging in 4 units: Area S₃ = (√3/4) * 4² = (√3/4)*16 = 4√3 ≈ 4 * 1.732 ≈ 6.928.Next, the square. The area is side². Each side is 3 units, so S₄ = 3² = 9. Hmm, that's already bigger than the triangle's area? Wait, but according to this, the square has a larger area than the triangle? But that contradicts my initial thought. Wait, maybe I made a mistake here.Wait, let me check again. The perimeter is 12. For the square, each side is 12/4 = 3. Area is 3*3=9. For the triangle, each side is 12/3=4. Area is (√3/4)*4²= (√3/4)*16=4√3≈6.928. So according to this, the square has a larger area than the triangle. Then the hexagon?Regular hexagon. Each side is 12/6=2 units. The area of a regular hexagon can be calculated using the formula (3√3/2) * side². So plugging in 2: Area S₆ = (3√3/2)*2² = (3√3/2)*4 = 6√3 ≈ 6*1.732≈10.392. So comparing these approximate values: S₆≈10.392, S₄=9, S₃≈6.928. So S₆ > S₄ > S₃. Which would be option B. But wait, the answer is supposed to be different? Wait, maybe my assumption of perimeter 12 is okay? Wait, but let me see. Maybe the perimeter is arbitrary. Let me try with a different perimeter to check.Suppose the perimeter is P. Let's denote the side lengths as follows: for triangle, a = P/3; for square, b = P/4; for hexagon, c = P/6.Then, area of triangle S₃ = (√3/4)a² = (√3/4)(P/3)² = (√3/4)(P²/9) = (√3 P²)/36.Area of square S₄ = b² = (P/4)² = P²/16.Area of hexagon S₆ = (3√3/2)c² = (3√3/2)(P/6)² = (3√3/2)(P²/36) = (3√3 P²)/72 = (√3 P²)/24.So now, let's compare S₃, S₄, S₆ in terms of coefficients:S₃: (√3)/36 ≈ 1.732 / 36 ≈ 0.0481 P²S₄: 1/16 ≈ 0.0625 P²S₆: (√3)/24 ≈ 1.732 / 24 ≈ 0.0721 P²So, indeed, S₆ ≈0.0721 P² > S₄≈0.0625 P² > S₃≈0.0481 P². Therefore, the correct order is S₆ > S₄ > S₃, which is option B. Wait, but in the problem statement, the options are:(A) S₃ > S₄ > S₆;(B) S₆ > S₄ > S₃;(C) S₆ > S₃ > S₄;(D) S₃ > S₆ > S₄;(E) S₄ > S₆ > S₃.So, according to the calculation, it's option B. But the initial thought was maybe different. Wait, when I first thought about the number of sides, I thought more sides mean more area, which is correct here. So the hexagon (6 sides) has the largest area, then square (4), then triangle (3). So the answer should be B.But wait, in the problem statement, the options are given, and according to my calculation, B is correct. However, sometimes there can be a mistake. Let me check the formulas again.Area of equilateral triangle: (√3/4) * side². Correct.Area of square: side². Correct.Area of regular hexagon: (3√3/2) * side². Correct. Since a regular hexagon can be divided into six equilateral triangles. Each triangle has area (√3/4) side², so six of them would be (6√3/4) side² = (3√3/2) side². So that formula is right.So, substituting the side lengths in terms of perimeter:Yes, for perimeter P, sides are P/3, P/4, P/6.Calculating coefficients:S₃: (√3)/4 * (P/3)² = √3/4 * P²/9 = √3 P² /36 ≈ 0.0481 P²S₄: (P/4)² = P²/16 ≈0.0625 P²S₆: (3√3/2)*(P/6)² = 3√3/2 * P²/36 = (3√3 P²)/(72) = √3 P² /24 ≈0.0721 P²Therefore, S₆ > S₄ > S₃, so answer is B. But wait, hold on, in my initial example with perimeter 12, the areas were approximately 6.928 (triangle), 9 (square), and 10.392 (hexagon). So S₆ > S₄ > S₃. So why does the problem even give different options? Is there a chance that the answer is different? Wait, maybe I made a miscalculation?Wait, let me check once again.For the regular hexagon: formula is (3√3/2) * side². For side length 2 (perimeter 12), area is (3√3/2)*4 = 6√3 ≈10.392. That's correct.For the square: side 3, area 9. Correct.For the triangle: side 4, area (√3/4)*16=4√3≈6.928. Correct.So in this case, the hexagon is the largest, then the square, then the triangle. Therefore, the answer is B. So why am I confused? Because maybe my initial thought was conflicting? Wait, no. Wait, in some sources, I remember that among regular polygons with the same perimeter, the one with more sides has a larger area, so indeed hexagon should be bigger than square, which is bigger than triangle. So that's correct. So the answer should be B.But looking at the options again: (B) S₆ > S₄ > S₃. So that's correct. Therefore, the correct answer is B. Wait, but in the problem statement, the answer given in the user's mind is different? Wait, no. Wait, the user hasn't provided any prior answer. Wait, the user is just posing the problem. So perhaps the answer is B. But let me check once again to be absolutely sure.Alternatively, maybe the problem is in Chinese or another language, and the translation is slightly different? Wait, the problem states: "For a regular triangle, a square, and a regular hexagon with the same perimeter, their areas are S3, S4, S6 respectively..." So same perimeter. Then, order of areas. The answer should be B.But let me check once again the general formula. The area of a regular polygon with n sides and perimeter P is (P² / (4n tan(π/n)) ). Let me verify this formula. Yes, the area A of a regular polygon with n sides of length s is (n * s²) / (4 tan(π/n)). Since perimeter P = n*s, so s = P/n. Plugging back in: A = n*(P/n)² / (4 tan(π/n)) ) = (P² / (4n tan(π/n)) ). So that's correct.Therefore, for each n (3,4,6), compute the area as P²/(4n tan(π/n)).Therefore, the coefficient for each is 1/(4n tan(π/n)).So let's compute these coefficients:For n=3 (triangle):tan(π/3) = √3 ≈1.732, so coefficient is 1/(4*3*1.732) ≈1/(12*1.732)≈1/20.784≈0.0481, which matches previous S₃.For n=4 (square):tan(π/4)=1, coefficient 1/(4*4*1)=1/16≈0.0625, which matches S₄.For n=6 (hexagon):tan(π/6)=1/√3≈0.577, coefficient 1/(4*6*0.577)≈1/(24*0.577)≈1/13.856≈0.0722, which matches S₆≈0.0721.Therefore, coefficients are increasing from n=3 to n=4 to n=6. Wait, but n=6 is larger than n=4. Wait, but the coefficient for n=6 is 0.0722, which is higher than n=4's 0.0625, which is higher than n=3's 0.0481. Therefore, S₆ > S₄ > S₃, so answer is B. Therefore, the correct answer is B. So why is the system expecting maybe another answer? Wait, maybe the problem was presented differently? Wait, no, according to the problem, perimeter is same. Therefore, answer must be B.Alternatively, maybe the user made a mistake in the problem statement? Let me check again. The problem says: "For a regular triangle, a square, and a regular hexagon with the same perimeter, their areas are S_{3}, S_{4}, S_{6} respectively, then (A) S₃>S₄>S₆; (B) S₆>S₄>S₃; (C) S₆>S₃>S₄; (D) S₃>S₆>S₄; (E) S₄>S₆>S₃."No, problem statement is correct. So according to calculations, B is correct. But in some sources, I recall that for given perimeter, the regular polygon with more sides encloses greater area. So hexagon > square > triangle. Therefore, S₆ > S₄ > S₃. So answer is B. So why is the option B present? Then, the answer must be B. Therefore, unless there's a miscalculation, which I have checked multiple times.Alternatively, maybe the original problem refers to "regular hexagon" as something different? No, regular hexagon is standard. So the answer is B. Therefore, I think the answer is B. But wait, the options given in the problem include (C) S₆ > S₃ > S₄. Wait, maybe I miscalculated the coefficients?Wait, let's recast the coefficients:Using the formula A = P²/(4n tan(π/n)), so for each shape:For triangle (n=3):A = P²/(4*3*tan(60°)) = P²/(12*√3) ≈ P²/(20.7846) ≈0.0481 P².For square (n=4):A = P²/(4*4*tan(45°)) = P²/(16*1) = P²/16 ≈0.0625 P².For hexagon (n=6):A = P²/(4*6*tan(30°)) = P²/(24*(1/√3)) = P²/(24/1.732) ≈ P²/13.856 ≈0.0722 P².So, yes, S₆ > S₄ > S₃. Therefore, answer is B.Alternatively, perhaps someone thought that the area of the hexagon is less than the square? But according to the numbers, with perimeter 12, hexagon area ≈10.392, square 9, triangle≈6.928. So clearly, hexagon is largest. Therefore, answer B.Alternatively, maybe the problem was stated differently in terms of same side length instead of same perimeter? But the problem clearly states same perimeter.Therefore, I think the answer is B. But let me check once again. Suppose all have the same perimeter. For regular polygons with the same perimeter, the one with more sides has a larger area. So 6 >4 >3, therefore S₆ > S₄ > S₃. So answer is B.Therefore, I think the answer is B.Final Answerboxed{B}</think>Given a regular triangle, a square, and a regular hexagon with the same perimeter, we need to compare their areas ( S_3 ), ( S_4 ), and ( S_6 ).1. Perimeter Consideration: - Let the common perimeter be ( P ). - The side lengths for each shape are: - Triangle: ( frac{P}{3} ) - Square: ( frac{P}{4} ) - Hexagon: ( frac{P}{6} )2. Area Formulas: - Regular Triangle (Equilateral): [ S_3 = frac{sqrt{3}}{4} left( frac{P}{3} right)^2 = frac{sqrt{3} P^2}{36} ] - Square: [ S_4 = left( frac{P}{4} right)^2 = frac{P^2}{16} ] - Regular Hexagon: [ S_6 = frac{3sqrt{3}}{2} left( frac{P}{6} right)^2 = frac{sqrt{3} P^2}{24} ]3. Comparing Coefficients: - Convert the areas to coefficients of ( P^2 ): - ( S_3 approx frac{1.732}{36} P^2 approx 0.0481 P^2 ) - ( S_4 approx frac{1}{16} P^2 approx 0.0625 P^2 ) - ( S_6 approx frac{1.732}{24} P^2 approx 0.0721 P^2 )4. Conclusion: - The coefficients show that ( S_6 > S_4 > S_3 ).Thus, the correct answer is (boxed{B}).,<think>Alright, so I need to figure out which of the options (A) through (E) is correct when comparing the areas of a regular triangle, a square, and a regular hexagon that all have the same perimeter. The areas are denoted as S₃, S₄, and S₆ respectively. Let me start by recalling some basic formulas for regular polygons.First, the perimeter of a regular polygon is the number of sides multiplied by the length of one side. Since all three shapes have the same perimeter, I can denote the perimeter as P. Then, for each shape, the side length will be P divided by the number of sides. Let me write that down:- For the regular triangle (3 sides), each side length is a₃ = P / 3.- For the square (4 sides), each side length is a₄ = P / 4.- For the regular hexagon (6 sides), each side length is a₆ = P / 6.Next, I need to find the area of each regular polygon. The formula for the area of a regular polygon with n sides of length a is:Area = (n * a²) / (4 * tan(π/n))Alternatively, I can also use the formula involving the apothem, but maybe this formula is more straightforward here. Let me verify this formula. Yes, for regular polygons, the area can be calculated using that formula. So I think that's correct.So, let's apply this formula to each shape. Let's start with the triangle (n=3):S₃ = (3 * a₃²) / (4 * tan(π/3))Similarly, for the square (n=4):S₄ = (4 * a₄²) / (4 * tan(π/4)) = (a₄²) / tan(π/4)And for the hexagon (n=6):S₆ = (6 * a₆²) / (4 * tan(π/6)) = (3 * a₆²) / (2 * tan(π/6))But since all perimeters are the same, P = 3a₃ = 4a₄ = 6a₆. So I can express all side lengths in terms of P. Let me express each area in terms of P.Starting with the triangle:a₃ = P / 3So,S₃ = [3 * (P/3)²] / [4 * tan(π/3)] = [3 * P² / 9] / [4 * tan(π/3)] = [P² / 3] / [4 * tan(π/3)] = P² / [12 * tan(π/3)]Similarly, for the square:a₄ = P / 4So,S₄ = [4 * (P/4)²] / [4 * tan(π/4)] = [4 * P² / 16] / [4 * tan(π/4)] = [P² / 4] / [4 * tan(π/4)] = P² / [16 * tan(π/4)]Wait, hold on. Let me check that again. The formula is (n * a²)/(4 * tan(π/n)). So for the square, n=4:S₄ = (4 * (P/4)²) / (4 * tan(π/4)) = (4 * P²/16) / (4 * tan(π/4)) = (P²/4) / (4 * tan(π/4)) = P² / (16 * tan(π/4))Yes, that's correct.Now for the hexagon:a₆ = P / 6S₆ = (6 * (P/6)²) / (4 * tan(π/6)) = (6 * P²/36) / (4 * tan(π/6)) = (P²/6) / (4 * tan(π/6)) = P² / (24 * tan(π/6))So now, all areas are expressed in terms of P² divided by some constants. Since P is the same for all, the area will be inversely proportional to those denominators. So to compare S₃, S₄, S₆, we can compare the denominators: the smaller the denominator, the larger the area.Therefore, the order of areas will be the reverse order of their denominators.So let's compute each denominator:Denominator for S₃: 12 * tan(π/3)Denominator for S₄: 16 * tan(π/4)Denominator for S₆: 24 * tan(π/6)We need to compute these values numerically to compare them.First, compute each tangent:tan(π/3): π/3 radians is 60 degrees. tan(60°) = √3 ≈ 1.732tan(π/4): π/4 radians is 45 degrees. tan(45°) = 1tan(π/6): π/6 radians is 30 degrees. tan(30°) = 1/√3 ≈ 0.577Therefore:Denominator S₃: 12 * √3 ≈ 12 * 1.732 ≈ 20.784Denominator S₄: 16 * 1 = 16Denominator S₆: 24 * (1/√3) ≈ 24 * 0.577 ≈ 13.856So the denominators from smallest to largest:S₆: ≈13.856 (smallest denominator, largest area)S₄: 16S₃: ≈20.784 (largest denominator, smallest area)Therefore, the areas should be ordered as S₆ > S₄ > S₃. But wait, that seems conflicting with the answer options. Let me check again.Wait, no. Wait, the area is P² divided by the denominator. So if the denominator is smallest for S₆, then S₆ is the largest. Then S₄ with denominator 16, so next. Then S₃ with denominator ~20.784, so smallest. So areas: S₆ > S₄ > S₃. Which would be option (B). But let me check the answer options again.Options:(A) S₃>S₄>S₆(B) S₆>S₄>S₃(C) S₆>S₃>S₄(D) S₃>S₆>S₄(E) S₄>S₆>S₃So according to my calculation, the order is S₆ > S₄ > S₃, which is option (B). But wait, that seems counterintuitive because I thought regular polygons with more sides have larger areas. Hmm.Wait, actually, for a given perimeter, as the number of sides increases, the area increases. So a hexagon should have a larger area than a square, which in turn should have a larger area than a triangle. So S₆ > S₄ > S₃, which would be option (B). So that's correct? But let me check my calculations again.Wait, maybe I made a mistake in the denominators. Let's re-express each area.Wait, S₃ = P² / [12 * tan(π/3)] ≈ P² / (12 * 1.732) ≈ P² / 20.784S₄ = P² / [16 * tan(π/4)] = P² / (16 * 1) = P² / 16 ≈ P² / 16S₆ = P² / [24 * tan(π/6)] ≈ P² / (24 * 0.577) ≈ P² / 13.856Therefore, S₆ ≈ P² /13.856, which is the largest, S₄ ≈ P² /16, which is next, and S₃ ≈ P² /20.784, which is smallest. So yes, S₆ > S₄ > S₃. Therefore, option (B).But wait, when I first thought about it, I thought maybe the answer is different, but according to the calculations, it's (B). However, I recall that for fixed perimeter, the area increases with the number of sides, approaching a circle. So the hexagon (6 sides) should have the largest area, then square (4), then triangle (3). Therefore, the order S₆ > S₄ > S₃, which is option (B). So the answer is (B). However, let me check once again.Alternatively, maybe there's an error in the formula. Let me recheck the formula for area of regular polygon.The formula is (1/4) * n * s² * cot(π/n), where s is the side length. Yes, that's equivalent to what I used: (n * a²)/(4 * tan(π/n)), since cot(x) = 1/tan(x). So yes, cot(π/n) is 1/tan(π/n), so the formula is correct.Therefore, all steps are correct. So the area is inversely proportional to tan(π/n). So as n increases, tan(π/n) decreases, so 1/tan(π/n) increases, so the area increases. Therefore, for fixed perimeter, higher n leads to higher area. Therefore, hexagon (n=6) has the largest area, then square (n=4), then triangle (n=3). Therefore, S₆ > S₄ > S₃, which is option (B).But wait, wait. The user's answer options: (B) is S₆ > S₄ > S₃. So if my conclusion is (B), then that's the answer. But I just want to check if maybe I miscalculated the denominators.Alternatively, maybe I confused perimeter with something else. Wait, in the problem statement, it says "the same perimeter". So for example, for a triangle, square, and hexagon with the same perimeter, which one has the largest area? Since regular polygons with more sides have larger area for the same perimeter, the hexagon should be the largest, then square, then triangle, so S₆ > S₄ > S₃, which is (B). Therefore, the answer should be (B). However, I need to check once more.Alternatively, let's calculate each area numerically with a specific perimeter. Let's take P = 12 units, for simplicity. Then:For the triangle:a₃ = 12 / 3 = 4. Each side is 4. Area S₃ = (√3 / 4) * a² = (√3 / 4) * 16 = 4√3 ≈ 6.928.For the square:a₄ = 12 / 4 = 3. Area S₄ = a² = 9.For the hexagon:a₆ = 12 / 6 = 2. Area S₆ = (3√3 / 2) * a² = (3√3 / 2) * 4 = 6√3 ≈ 10.392.So comparing these numerical values:S₃ ≈ 6.928,S₄ = 9,S₆ ≈ 10.392.Therefore, S₆ > S₄ > S₃, which is option (B). So that's correct.Wait, but earlier, using the formula with cotangent, we got similar results. So if we take P = 12:S₃ = (3 * 4²) / (4 * tan(60°)) = (48) / (4 * 1.732) ≈ 48 / 6.928 ≈ 6.928.Similarly, S₄ = (4 * 3²) / (4 * tan(45°)) = (36) / (4 * 1) = 9.S₆ = (6 * 2²) / (4 * tan(30°)) = (24) / (4 * 0.577) ≈ 24 / 2.308 ≈ 10.392.Yes, same results. Therefore, the correct order is S₆ > S₄ > S₃, which is option (B). Therefore, the answer is (B).But wait, the problem states "their areas are S₃, S₄, S₆ respectively". So S₃ is the area of the triangle, S₄ of the square, S₆ of the hexagon. Therefore, the order is S₆ > S₄ > S₃, which is option (B). However, in the initial answer options, (B) is S6 > S4 > S3, which matches. So (B) is the correct answer. Therefore, the answer is boxed{B}.Final Answerboxed{B}</think>Given a regular triangle, a square, and a regular hexagon with the same perimeter, we need to compare their areas ( S_3 ), ( S_4 ), and ( S_6 ) respectively.1. Perimeter and Side Lengths: - Perimeter ( P ) is the same for all shapes. - Side lengths: - Triangle: ( a_3 = frac{P}{3} ) - Square: ( a_4 = frac{P}{4} ) - Hexagon: ( a_6 = frac{P}{6} )2. Area Formulas: - Area of a regular polygon with ( n ) sides: ( text{Area} = frac{n cdot a^2}{4 cdot tan(pi/n)} )3. Calculating Areas: - Triangle (( n = 3 )): [ S_3 = frac{3 cdot left(frac{P}{3}right)^2}{4 cdot tanleft(frac{pi}{3}right)} = frac{P^2}{12 cdot tanleft(frac{pi}{3}right)} = frac{P^2}{12 cdot sqrt{3}} ] - Square (( n = 4 )): [ S_4 = frac{4 cdot left(frac{P}{4}right)^2}{4 cdot tanleft(frac{pi}{4}right)} = frac{P^2}{16 cdot tanleft(frac{pi}{4}right)} = frac{P^2}{16} ] - Hexagon (( n = 6 )): [ S_6 = frac{6 cdot left(frac{P}{6}right)^2}{4 cdot tanleft(frac{pi}{6}right)} = frac{P^2}{24 cdot tanleft(frac{pi}{6}right)} = frac{P^2}{24 cdot frac{1}{sqrt{3}}} ]4. Comparing Denominators: - Denominators (proportional to the inverse of the area): - ( 12 cdot sqrt{3} approx 20.78 ) - ( 16 ) - ( 24 cdot frac{1}{sqrt{3}} approx 13.86 ) - Smaller denominators correspond to larger areas. Therefore, the order of areas is ( S_6 > S_4 > S_3 ).5. Numerical Verification: - Using ( P = 12 ): - Triangle: ( S_3 approx 6.928 ) - Square: ( S_4 = 9 ) - Hexagon: ( S_6 approx 10.392 )Thus, the correct order is ( S_6 > S_4 > S_3 ), corresponding to option (B).[boxed{B}]]

🔑:3. (B)Let the same perimeter be 12 l(l>0), then their areas arebegin{array}{l}S_{3}=frac{1}{2} times 4 l times 4 l times sin 60^{circ}=4 sqrt{3} l^{2}, S_{4}=3 l times 3 l=9 l^{2}, S_{6}=6 sqrt{3} l^{2} . therefore quad S_{6}>S_{4}>S_{3} .end{array}