💡Okay, so I have this problem where a segment PQ is divided into several smaller segments, and on each of these smaller segments, a square is constructed. The question is asking for the length of the path along the arrows if the length of PQ is 73. I need to figure out how to calculate this path length.First, I should visualize what's happening here. If PQ is divided into smaller segments, each of these segments becomes the side of a square. So, if I imagine the segment PQ, it's like it's been split into, say, n smaller equal parts, and on each part, a square is built outward. Now, the path along the arrows probably means moving from one square to the next, following some specific direction indicated by the arrows.Since the problem mentions moving along the arrows, I think this path isn't just the straight line of PQ but rather a more complex route that goes around the squares. Maybe it's a zigzag path or something similar. But without seeing the figure, I have to make some assumptions.Let me think about how the path could be constructed. If each square is built on a smaller segment of PQ, then each square has a side length equal to that smaller segment. If the entire PQ is 73 units long and it's divided into, for example, k smaller segments, each smaller segment would have a length of 73/k.Now, if a square is constructed on each of these smaller segments, each square would have sides of length 73/k. The path along the arrows might involve moving along some combination of these squares. Perhaps it's moving along the perimeter of each square or moving from one square to the next in a specific pattern.Wait, the problem says "the length of the path along the arrows." So, maybe the arrows indicate a specific traversal path that goes through each square. If I consider that each square contributes a certain amount to the total path length, I need to figure out how much each square adds.If the path goes along the arrows, maybe it's moving along three sides of each square instead of just one. For example, if you have a square, and you move along three of its sides, that would be three times the side length. So, if each square has a side length of 73/k, then moving along three sides would be 3*(73/k).But if there are k squares, each contributing 3*(73/k) to the total path, then the total length would be k*(3*(73/k)) = 3*73 = 219.Wait, that seems too straightforward. Let me check my reasoning. If each square contributes three sides to the path, and there are k squares, each with side length 73/k, then the total path length would indeed be 3*73 = 219, regardless of the number of divisions k. That's interesting because it means the total path length is always three times the original segment PQ, no matter how it's divided.But is that accurate? Let me think about a simple case where PQ is divided into two equal parts. Each part would be 73/2 = 36.5. On each of these, a square is constructed. If the path goes along three sides of each square, then for each square, the path would be 3*36.5 = 109.5. Since there are two squares, the total path length would be 2*109.5 = 219. That matches my earlier calculation.What if PQ is divided into three equal parts? Each part would be 73/3 ≈ 24.333. Each square would have sides of 24.333, and moving along three sides would be 3*24.333 ≈ 73. So, with three squares, the total path length would be 3*73 ≈ 219. Again, the same result.This seems consistent. No matter how many segments PQ is divided into, the total path length along the arrows is always three times the length of PQ. Therefore, if PQ is 73, the total path length is 3*73 = 219.I think this makes sense because each smaller segment contributes three times its length to the path, and since the sum of all smaller segments is 73, the total path is 3*73. So, the answer should be 219.

💡First, I need to determine the number that, when multiplied by 9999, results in 4691110842.To find this number, I will divide 4691110842 by 9999.Performing the division:4691110842 ÷ 9999 = 469211Thus, the number is 469211.

💡Alright, so I have this problem here: I need to find the smallest value of ( n ) such that the sum ( S_n = 9 + 19 + 29 + 39 + cdots + a_n ) is greater than ( 10^5 ). Okay, let's break this down step by step.First, I recognize that this is an arithmetic sequence because each term increases by 10. The first term ( a_1 ) is 9, and the common difference ( d ) is 10. So, the sequence is 9, 19, 29, 39, and so on.Now, I need to find the sum of the first ( n ) terms of this sequence. The formula for the sum of the first ( n ) terms of an arithmetic sequence is:[S_n = frac{n}{2} times (a_1 + a_n)]But wait, I don't know ( a_n ) yet. I need to find the expression for the ( n )-th term. The formula for the ( n )-th term of an arithmetic sequence is:[a_n = a_1 + (n - 1) times d]Plugging in the values I have:[a_n = 9 + (n - 1) times 10][a_n = 9 + 10n - 10][a_n = 10n - 1]Okay, so the ( n )-th term is ( 10n - 1 ). Now, let's plug this back into the sum formula:[S_n = frac{n}{2} times (9 + (10n - 1))][S_n = frac{n}{2} times (10n + 8)][S_n = frac{n}{2} times 10n + frac{n}{2} times 8][S_n = 5n^2 + 4n]So, the sum ( S_n ) is equal to ( 5n^2 + 4n ). Now, I need to find the smallest integer ( n ) such that ( S_n > 10^5 ). That means I need to solve the inequality:[5n^2 + 4n > 100,000]Hmm, this is a quadratic inequality. To solve it, I can first set the equation equal to 100,000 and solve for ( n ):[5n^2 + 4n = 100,000]Let me rearrange this equation to standard quadratic form:[5n^2 + 4n - 100,000 = 0]Now, I can use the quadratic formula to solve for ( n ). The quadratic formula is:[n = frac{-b pm sqrt{b^2 - 4ac}}{2a}]In this equation, ( a = 5 ), ( b = 4 ), and ( c = -100,000 ). Plugging these values into the formula:[n = frac{-4 pm sqrt{4^2 - 4 times 5 times (-100,000)}}{2 times 5}][n = frac{-4 pm sqrt{16 + 2,000,000}}{10}][n = frac{-4 pm sqrt{2,000,016}}{10}]Now, I need to calculate the square root of 2,000,016. Let me see, ( sqrt{2,000,016} ) is approximately 1414.2136. So,[n = frac{-4 pm 1414.2136}{10}]Since ( n ) represents the number of terms, it must be positive. So, I'll take the positive root:[n = frac{-4 + 1414.2136}{10}][n = frac{1410.2136}{10}][n approx 141.02136]Since ( n ) must be an integer, and we need ( S_n ) to be greater than 100,000, we round up to the next whole number. So, ( n = 142 ).But wait, let me double-check this. If I plug ( n = 141 ) into the sum formula:[S_{141} = 5(141)^2 + 4(141)][S_{141} = 5(19881) + 564][S_{141} = 99,405 + 564][S_{141} = 99,969]That's just under 100,000. Now, let's try ( n = 142 ):[S_{142} = 5(142)^2 + 4(142)][S_{142} = 5(20164) + 568][S_{142} = 100,820 + 568][S_{142} = 101,388]Okay, that's definitely over 100,000. So, ( n = 142 ) is indeed the smallest integer where the sum exceeds 100,000.I think that's it. I went through the steps, checked my work, and made sure the answer makes sense. It looks like 142 is the correct answer.

💡Okay, so I have this function f(x) = 4^(x+1)/(4^x + 2), and I need to find S_n, which is the sum of f(1/n) + f(2/n) + ... + f((n-1)/n). Hmm, that seems a bit complicated at first glance, but maybe I can find a pattern or a property of the function that can help simplify the sum.Let me write down the function again to make sure I have it right: f(x) = 4^(x+1)/(4^x + 2). I wonder if there's a way to simplify this expression. Maybe I can factor out something from the numerator or the denominator.Looking at the numerator, 4^(x+1) is the same as 4 * 4^x. So, I can rewrite f(x) as (4 * 4^x)/(4^x + 2). That might be helpful. So, f(x) = 4 * 4^x / (4^x + 2). Hmm, maybe I can factor out 4^x from the denominator? Wait, the denominator is 4^x + 2, so I can't factor out 4^x directly. Maybe I can divide numerator and denominator by 4^x? Let's try that.If I divide numerator and denominator by 4^x, I get f(x) = 4 / (1 + 2 / 4^x). That simplifies to 4 / (1 + 2 * 4^(-x)). Hmm, that might be useful later on. I'll keep that in mind.Now, the problem is asking for S_n = f(1/n) + f(2/n) + ... + f((n-1)/n). So, it's a sum of f evaluated at points that are fractions with denominator n, starting from 1/n up to (n-1)/n. I wonder if there's a symmetry in the function f(x) that can help me pair terms in the sum and simplify it.Let me consider f(x) + f(1 - x). Maybe this sum has a nice property. Let's compute f(1 - x):f(1 - x) = 4^( (1 - x) + 1 ) / (4^(1 - x) + 2) = 4^(2 - x) / (4^(1 - x) + 2).Hmm, let's simplify that. 4^(2 - x) is the same as 4^2 * 4^(-x) = 16 / 4^x. Similarly, 4^(1 - x) is 4 / 4^x. So, substituting back, f(1 - x) = (16 / 4^x) / (4 / 4^x + 2).Let me simplify the denominator: 4 / 4^x + 2 = 4 * 4^(-x) + 2. So, f(1 - x) = (16 / 4^x) / (4 * 4^(-x) + 2).Wait, let me write everything in terms of 4^x to make it easier. Let me denote y = 4^x. Then, 4^(-x) = 1/y. So, substituting, f(1 - x) becomes (16 / y) / (4 / y + 2) = (16 / y) / ( (4 + 2y)/y ) = (16 / y) * (y / (4 + 2y)) ) = 16 / (4 + 2y).Simplify that: 16 / (4 + 2y) = (16) / (2*(2 + y)) ) = 8 / (2 + y). But y = 4^x, so f(1 - x) = 8 / (2 + 4^x).Now, let's recall that f(x) was 4 * 4^x / (4^x + 2). So, f(x) = 4y / (y + 2). And f(1 - x) = 8 / (2 + y). Let's add these two together:f(x) + f(1 - x) = (4y / (y + 2)) + (8 / (2 + y)).Since the denominators are the same, we can combine them:= (4y + 8) / (y + 2) = 4(y + 2) / (y + 2) = 4.Wow, that's nice! So, f(x) + f(1 - x) = 4. That seems like a key insight here.So, for any x, f(x) + f(1 - x) = 4. That means if I have a term f(k/n) in the sum S_n, I can pair it with f(1 - k/n) = f((n - k)/n), and their sum is 4.Now, looking back at S_n, it's the sum from k = 1 to k = n - 1 of f(k/n). So, the terms are f(1/n), f(2/n), ..., f((n - 1)/n). If I pair the first term with the last term, the second term with the second last term, and so on, each pair should add up to 4.How many such pairs are there? Well, since there are n - 1 terms in total, if n - 1 is even, then we can pair all terms perfectly. If n - 1 is odd, there will be a middle term that doesn't have a pair. Wait, but n is starting from 2, so n - 1 is at least 1. Let me think.Wait, n is given as 2, 3, ..., so n - 1 is 1, 2, 3, etc. So, when n is even, n - 1 is odd, and when n is odd, n - 1 is even. Hmm, but regardless, the number of pairs is (n - 1)/2 if n - 1 is odd, but actually, wait, no. Let me think again.Wait, actually, the number of pairs is floor((n - 1)/2). But since we're pairing terms from both ends, the number of pairs is (n - 1)/2 when n is odd, and (n - 2)/2 when n is even? Wait, no, that might not be the right way to think about it.Wait, perhaps a better approach is to note that for each k from 1 to n - 1, the term f(k/n) pairs with f((n - k)/n). So, each pair is f(k/n) + f((n - k)/n) = 4. How many such unique pairs are there?Well, if n is even, say n = 2m, then n - 1 = 2m - 1, which is odd. So, the number of pairs would be m - 1, and there's a middle term when k = m, which is f(m/n) = f(1/2). But wait, in our case, n starts from 2, so when n is even, n - 1 is odd, so there is a middle term. Similarly, when n is odd, n - 1 is even, so all terms can be paired without a middle term.Wait, but in our sum S_n, we have terms from k = 1 to k = n - 1. So, if n is even, say n = 4, then the terms are f(1/4), f(2/4), f(3/4). So, pairing f(1/4) with f(3/4), which is 4, and f(2/4) is f(1/2), which is a middle term. So, in this case, S_n would be 4 + f(1/2).But wait, earlier we saw that f(x) + f(1 - x) = 4, so f(1/2) + f(1 - 1/2) = f(1/2) + f(1/2) = 2f(1/2) = 4, so f(1/2) = 2. So, in the case of n = 4, S_n = 4 + 2 = 6. But according to the formula we might be deriving, if n = 4, S_n should be 2n - 2 = 8 - 2 = 6. That matches.Similarly, let's test n = 3. Then, S_n = f(1/3) + f(2/3). Pairing f(1/3) and f(2/3), which sum to 4. So, S_n = 4. According to the formula, 2n - 2 = 6 - 2 = 4. That also matches.Wait, so maybe regardless of whether n is even or odd, S_n = 2n - 2. Let me check n = 2. Then, S_n = f(1/2). As we saw earlier, f(1/2) = 2. So, S_n = 2. According to the formula, 2n - 2 = 4 - 2 = 2. That matches too.So, it seems that regardless of n, S_n = 2n - 2. But let me try to see why that is the case.Since each pair f(k/n) + f((n - k)/n) = 4, and the number of such pairs is (n - 1)/2 when n is odd, but when n is even, we have (n - 2)/2 pairs and one middle term f(1/2) = 2. So, let's compute S_n in both cases.Case 1: n is odd. Then, n - 1 is even, so the number of pairs is (n - 1)/2. Each pair sums to 4, so S_n = 4 * (n - 1)/2 = 2(n - 1) = 2n - 2.Case 2: n is even. Then, n - 1 is odd, so the number of pairs is (n - 2)/2, and there's one middle term f(1/2) = 2. So, S_n = 4 * (n - 2)/2 + 2 = 2(n - 2) + 2 = 2n - 4 + 2 = 2n - 2.So, in both cases, whether n is odd or even, S_n = 2n - 2. Therefore, the answer is S_n = 2n - 2.I think that makes sense. The key was recognizing the symmetry in the function f(x) + f(1 - x) = 4, which allowed us to pair terms in the sum and simplify it significantly. By considering both cases where n is odd and even, we confirmed that the formula holds in general. So, I'm confident that S_n = 2n - 2 is the correct answer.

💡Alright, so I have this problem here: Given ( U = {1, 2, a^2 + 2a - 3} ), ( A = {|a - 2|, 2} ), and ( C_U A = {0} ), what is the value of ( a )?The options are:A: -3 or 1 B: 2 C: 3 or 1 D: 1Okay, let's break this down step by step. I'm a bit rusty on set theory, but I'll try to recall.First, ( U ) is the universal set, which contains elements ( 1 ), ( 2 ), and ( a^2 + 2a - 3 ). So, ( U ) has three elements, one of which depends on ( a ).Then, ( A ) is a subset of ( U ), containing ( |a - 2| ) and ( 2 ). So, ( A ) has two elements, one of which is the absolute value of ( a - 2 ).Now, ( C_U A ) is the complement of ( A ) with respect to ( U ), and it's given as ( {0} ). That means all elements in ( U ) that are not in ( A ) are just ( 0 ).So, the complement of ( A ) in ( U ) is ( {0} ). That tells me that ( U ) must contain ( 0 ), because the complement is non-empty. Therefore, ( a^2 + 2a - 3 ) must be equal to ( 0 ). Let me write that down:( a^2 + 2a - 3 = 0 )This is a quadratic equation. To solve for ( a ), I can factor it or use the quadratic formula. Let me try factoring first.Looking for two numbers that multiply to ( -3 ) and add up to ( 2 ). Hmm, 3 and -1? Yes, because ( 3 times (-1) = -3 ) and ( 3 + (-1) = 2 ). So, the equation factors as:( (a + 3)(a - 1) = 0 )Setting each factor equal to zero gives:( a + 3 = 0 ) ⇒ ( a = -3 )or( a - 1 = 0 ) ⇒ ( a = 1 )So, possible values for ( a ) are ( -3 ) and ( 1 ). But I need to check if both these values satisfy the original condition that ( C_U A = {0} ).Let's check each value of ( a ):**Case 1: ( a = -3 )**Substitute ( a = -3 ) into ( U ):( U = {1, 2, (-3)^2 + 2(-3) - 3} = {1, 2, 9 - 6 - 3} = {1, 2, 0} )Now, substitute ( a = -3 ) into ( A ):( A = {|-3 - 2|, 2} = {|-5|, 2} = {5, 2} )So, ( A = {5, 2} ) and ( U = {1, 2, 0} ). The complement of ( A ) in ( U ) would be ( U ) minus ( A ), which is ( {1, 0} ). But the problem states that ( C_U A = {0} ). Here, it's ( {1, 0} ), which doesn't match. So, ( a = -3 ) doesn't work.**Case 2: ( a = 1 )**Substitute ( a = 1 ) into ( U ):( U = {1, 2, (1)^2 + 2(1) - 3} = {1, 2, 1 + 2 - 3} = {1, 2, 0} )Now, substitute ( a = 1 ) into ( A ):( A = {|1 - 2|, 2} = {|-1|, 2} = {1, 2} )So, ( A = {1, 2} ) and ( U = {1, 2, 0} ). The complement of ( A ) in ( U ) is ( U ) minus ( A ), which is ( {0} ). This matches the given condition ( C_U A = {0} ). So, ( a = 1 ) works.Therefore, the only valid solution is ( a = 1 ).Wait, but the options include both -3 and 1 as possible answers in option A. But from my calculations, only ( a = 1 ) works. So, why is option A suggesting both -3 and 1?Let me double-check my work. Maybe I made a mistake when substituting ( a = -3 ).For ( a = -3 ):( U = {1, 2, (-3)^2 + 2(-3) - 3} = {1, 2, 9 - 6 - 3} = {1, 2, 0} )( A = {|-3 - 2|, 2} = {5, 2} )So, ( C_U A = U - A = {1, 0} ), which is not equal to ( {0} ). So, indeed, ( a = -3 ) doesn't satisfy the condition.Therefore, only ( a = 1 ) is valid. So, the correct answer should be D: 1.But wait, the initial quadratic equation gave me two solutions, but only one worked. So, in the options, A says -3 or 1, but only 1 is correct. So, why is A an option? Maybe the question is testing if I consider both solutions but realize only one works.Alternatively, perhaps I made a mistake in interpreting the complement. Let me think again.The complement of ( A ) in ( U ) is ( U ) minus ( A ). So, if ( A = {5, 2} ) and ( U = {1, 2, 0} ), then ( C_U A = {1, 0} ). But the problem says ( C_U A = {0} ). So, ( a = -3 ) is invalid.Hence, only ( a = 1 ) is correct.So, the answer should be D: 1.But let me see if there's another perspective. Maybe ( |a - 2| ) could be equal to 0? Let's see.If ( |a - 2| = 0 ), then ( a = 2 ). But ( a = 2 ) would make ( U = {1, 2, (2)^2 + 2(2) - 3} = {1, 2, 4 + 4 - 3} = {1, 2, 5} ). Then, ( A = {|2 - 2|, 2} = {0, 2} ). So, ( C_U A = U - A = {1, 5} ), which is not ( {0} ). So, ( a = 2 ) is also invalid.Therefore, the only valid solution is ( a = 1 ).I think I'm confident now that the correct answer is D: 1.

💡Okay, so I have this problem where I need to find the perimeter of triangle ABC. The points A, B, and C are given in polar coordinates: A is (4, θ₁), B is (5, θ₂), and C is (7, θ₃). The angles have differences: θ₁ - θ₂ = π/3 and θ₂ - θ₃ = π/3. Hmm, so that means the angles are each separated by 60 degrees. Interesting.First, I remember that in polar coordinates, a point is represented as (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. To find the distances between points A, B, and C, I probably need to convert these polar coordinates into Cartesian coordinates because it's easier to calculate distances in Cartesian form.So, let me recall the conversion formulas:- x = r * cos(θ)- y = r * sin(θ)Therefore, for point A:x_A = 4 * cos(θ₁)y_A = 4 * sin(θ₁)For point B:x_B = 5 * cos(θ₂)y_B = 5 * sin(θ₂)For point C:x_C = 7 * cos(θ₃)y_C = 7 * sin(θ₃)Now, since θ₁ - θ₂ = π/3 and θ₂ - θ₃ = π/3, that means θ₁ = θ₂ + π/3 and θ₂ = θ₃ + π/3. So, θ₁ is θ₃ + 2π/3. Maybe I can express all angles in terms of θ₃ to simplify things? Let me try that.Let’s set θ₃ as θ. Then θ₂ = θ + π/3 and θ₁ = θ + 2π/3.So, now I can write the coordinates in terms of θ:- A: (4cos(θ + 2π/3), 4sin(θ + 2π/3))- B: (5cos(θ + π/3), 5sin(θ + π/3))- C: (7cosθ, 7sinθ)Hmm, this might make it easier because now all points are expressed in terms of a single angle θ. Maybe I can compute the distances AB, BC, and CA using these expressions.Let me recall the distance formula between two points (x1, y1) and (x2, y2):Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]So, I need to compute AB, BC, and CA.Starting with AB:A is (4cos(θ + 2π/3), 4sin(θ + 2π/3))B is (5cos(θ + π/3), 5sin(θ + π/3))So, AB = sqrt[(5cos(θ + π/3) - 4cos(θ + 2π/3))^2 + (5sin(θ + π/3) - 4sin(θ + 2π/3))^2]This looks a bit complicated, but maybe I can simplify the trigonometric expressions.I remember that cos(A + B) and sin(A + B) can be expanded using angle addition formulas:cos(A + B) = cosA cosB - sinA sinBsin(A + B) = sinA cosB + cosA sinBLet me compute cos(θ + π/3) and sin(θ + π/3):cos(θ + π/3) = cosθ cos(π/3) - sinθ sin(π/3) = (1/2)cosθ - (√3/2)sinθsin(θ + π/3) = sinθ cos(π/3) + cosθ sin(π/3) = (1/2)sinθ + (√3/2)cosθSimilarly, cos(θ + 2π/3) and sin(θ + 2π/3):cos(θ + 2π/3) = cosθ cos(2π/3) - sinθ sin(2π/3) = (-1/2)cosθ - (√3/2)sinθsin(θ + 2π/3) = sinθ cos(2π/3) + cosθ sin(2π/3) = (-1/2)sinθ + (√3/2)cosθOkay, so plugging these back into the coordinates:For point A:x_A = 4 * [(-1/2)cosθ - (√3/2)sinθ] = -2cosθ - 2√3 sinθy_A = 4 * [(-1/2)sinθ + (√3/2)cosθ] = -2sinθ + 2√3 cosθFor point B:x_B = 5 * [(1/2)cosθ - (√3/2)sinθ] = (5/2)cosθ - (5√3/2)sinθy_B = 5 * [(1/2)sinθ + (√3/2)cosθ] = (5/2)sinθ + (5√3/2)cosθNow, let's compute the differences for AB:Δx = x_B - x_A = [(5/2)cosθ - (5√3/2)sinθ] - [-2cosθ - 2√3 sinθ]= (5/2)cosθ - (5√3/2)sinθ + 2cosθ + 2√3 sinθ= (5/2 + 2)cosθ + (-5√3/2 + 2√3)sinθ= (9/2)cosθ + (-5√3/2 + 4√3/2)sinθ= (9/2)cosθ + (-√3/2)sinθSimilarly, Δy = y_B - y_A = [(5/2)sinθ + (5√3/2)cosθ] - [-2sinθ + 2√3 cosθ]= (5/2)sinθ + (5√3/2)cosθ + 2sinθ - 2√3 cosθ= (5/2 + 2)sinθ + (5√3/2 - 2√3)cosθ= (9/2)sinθ + (5√3/2 - 4√3/2)cosθ= (9/2)sinθ + (√3/2)cosθSo, AB = sqrt[(9/2 cosθ - √3/2 sinθ)^2 + (9/2 sinθ + √3/2 cosθ)^2]Let me compute each squared term:First term squared:(9/2 cosθ - √3/2 sinθ)^2 = (81/4)cos²θ - (9√3)/2 cosθ sinθ + (3/4)sin²θSecond term squared:(9/2 sinθ + √3/2 cosθ)^2 = (81/4)sin²θ + (9√3)/2 sinθ cosθ + (3/4)cos²θAdding them together:(81/4 cos²θ + 3/4 sin²θ) + (81/4 sin²θ + 3/4 cos²θ) + (-9√3/2 cosθ sinθ + 9√3/2 cosθ sinθ)Notice that the cross terms (-9√3/2 and +9√3/2) cancel each other out.So, we have:81/4 cos²θ + 3/4 sin²θ + 81/4 sin²θ + 3/4 cos²θ= (81/4 + 3/4) cos²θ + (81/4 + 3/4) sin²θ= (84/4) cos²θ + (84/4) sin²θ= 21 cos²θ + 21 sin²θ= 21 (cos²θ + sin²θ)= 21 * 1= 21So, AB = sqrt(21)Wow, that simplified nicely! So, AB is sqrt(21). That's a constant, independent of θ. Interesting.Now, moving on to BC.Points B and C:B: (5cos(θ + π/3), 5sin(θ + π/3))C: (7cosθ, 7sinθ)So, BC = sqrt[(7cosθ - 5cos(θ + π/3))^2 + (7sinθ - 5sin(θ + π/3))^2]Again, let's use the angle addition formulas for cos(θ + π/3) and sin(θ + π/3), which we already computed earlier:cos(θ + π/3) = (1/2)cosθ - (√3/2)sinθsin(θ + π/3) = (1/2)sinθ + (√3/2)cosθSo, let's compute the differences:Δx = 7cosθ - 5[(1/2)cosθ - (√3/2)sinθ] = 7cosθ - (5/2)cosθ + (5√3/2)sinθ = (7 - 5/2)cosθ + (5√3/2)sinθ = (9/2)cosθ + (5√3/2)sinθΔy = 7sinθ - 5[(1/2)sinθ + (√3/2)cosθ] = 7sinθ - (5/2)sinθ - (5√3/2)cosθ = (7 - 5/2)sinθ - (5√3/2)cosθ = (9/2)sinθ - (5√3/2)cosθSo, BC = sqrt[(9/2 cosθ + 5√3/2 sinθ)^2 + (9/2 sinθ - 5√3/2 cosθ)^2]Let me compute each squared term:First term squared:(9/2 cosθ + 5√3/2 sinθ)^2 = (81/4)cos²θ + (45√3)/2 cosθ sinθ + (75/4)sin²θSecond term squared:(9/2 sinθ - 5√3/2 cosθ)^2 = (81/4)sin²θ - (45√3)/2 sinθ cosθ + (75/4)cos²θAdding them together:(81/4 cos²θ + 75/4 sin²θ) + (81/4 sin²θ + 75/4 cos²θ) + (45√3/2 cosθ sinθ - 45√3/2 cosθ sinθ)Again, the cross terms cancel out.So, we have:81/4 cos²θ + 75/4 sin²θ + 81/4 sin²θ + 75/4 cos²θ= (81/4 + 75/4) cos²θ + (75/4 + 81/4) sin²θ= (156/4) cos²θ + (156/4) sin²θ= 39 cos²θ + 39 sin²θ= 39 (cos²θ + sin²θ)= 39 * 1= 39So, BC = sqrt(39)Wow, that's another constant. So, BC is sqrt(39).Now, onto CA.Points C and A:C: (7cosθ, 7sinθ)A: (4cos(θ + 2π/3), 4sin(θ + 2π/3))So, CA = sqrt[(4cos(θ + 2π/3) - 7cosθ)^2 + (4sin(θ + 2π/3) - 7sinθ)^2]Again, let's use the angle addition formulas for cos(θ + 2π/3) and sin(θ + 2π/3), which we computed earlier:cos(θ + 2π/3) = (-1/2)cosθ - (√3/2)sinθsin(θ + 2π/3) = (-1/2)sinθ + (√3/2)cosθSo, let's compute the differences:Δx = 4[(-1/2)cosθ - (√3/2)sinθ] - 7cosθ = (-2cosθ - 2√3 sinθ) - 7cosθ = (-9cosθ - 2√3 sinθ)Δy = 4[(-1/2)sinθ + (√3/2)cosθ] - 7sinθ = (-2sinθ + 2√3 cosθ) - 7sinθ = (-9sinθ + 2√3 cosθ)So, CA = sqrt[(-9cosθ - 2√3 sinθ)^2 + (-9sinθ + 2√3 cosθ)^2]Let me compute each squared term:First term squared:(-9cosθ - 2√3 sinθ)^2 = 81cos²θ + 36√3 cosθ sinθ + 12 sin²θSecond term squared:(-9sinθ + 2√3 cosθ)^2 = 81sin²θ - 36√3 sinθ cosθ + 12 cos²θAdding them together:81cos²θ + 12 sin²θ + 81sin²θ + 12 cos²θ + (36√3 cosθ sinθ - 36√3 cosθ sinθ)Again, the cross terms cancel out.So, we have:81cos²θ + 12 sin²θ + 81sin²θ + 12 cos²θ= (81 + 12) cos²θ + (12 + 81) sin²θ= 93 cos²θ + 93 sin²θ= 93 (cos²θ + sin²θ)= 93 * 1= 93Thus, CA = sqrt(93)So, now we have all three sides:AB = sqrt(21)BC = sqrt(39)CA = sqrt(93)Therefore, the perimeter is sqrt(21) + sqrt(39) + sqrt(93)Wait, let me check if these square roots can be simplified further.sqrt(21) is already simplified.sqrt(39) is also simplified.sqrt(93) can be broken down: 93 = 3 * 31, both primes, so it's also simplified.So, the perimeter is sqrt(21) + sqrt(39) + sqrt(93). That's the exact value.Alternatively, if I want to approximate it numerically, I can calculate each square root:sqrt(21) ≈ 4.5837sqrt(39) ≈ 6.2450sqrt(93) ≈ 9.6437Adding them up: 4.5837 + 6.2450 + 9.6437 ≈ 20.4724But since the problem doesn't specify, I think the exact value is preferred, so I'll stick with sqrt(21) + sqrt(39) + sqrt(93).Wait, but let me double-check my calculations for CA because I might have made a mistake there.Looking back at CA:Δx = 4cos(θ + 2π/3) - 7cosθ = 4*(-1/2 cosθ - √3/2 sinθ) - 7cosθ = -2cosθ - 2√3 sinθ - 7cosθ = -9cosθ - 2√3 sinθΔy = 4sin(θ + 2π/3) - 7sinθ = 4*(-1/2 sinθ + √3/2 cosθ) - 7sinθ = -2sinθ + 2√3 cosθ - 7sinθ = -9sinθ + 2√3 cosθSo, squared terms:(-9cosθ - 2√3 sinθ)^2 = 81cos²θ + 36√3 cosθ sinθ + 12 sin²θ(-9sinθ + 2√3 cosθ)^2 = 81sin²θ - 36√3 sinθ cosθ + 12 cos²θAdding them:81cos²θ + 12 sin²θ + 81sin²θ + 12 cos²θ = 93cos²θ + 93 sin²θ = 93Yes, that seems correct. So, CA is indeed sqrt(93). So, I think my calculations are correct.Therefore, the perimeter is sqrt(21) + sqrt(39) + sqrt(93). I don't think there's a simpler form for this, so that should be the final answer.**Final Answer**The perimeter of triangle ( ABC ) is (boxed{sqrt{21} + sqrt{39} + sqrt{93}}).