💡Okay, so I have this problem where I need to find the probability of getting at least 10 heads when flipping a fair coin 12 times. Hmm, let me think about how to approach this.First, I know that each coin flip is an independent event with two possible outcomes: heads or tails. Since the coin is fair, the probability of getting heads (H) is 0.5, and the probability of getting tails (T) is also 0.5.I remember that when dealing with probabilities involving multiple trials and specific outcomes, the binomial probability formula is useful. The binomial formula is:P(k) = C(n, k) * p^k * (1-p)^(n-k)Where:- P(k) is the probability of getting exactly k successes (in this case, heads) in n trials.- C(n, k) is the combination of n things taken k at a time.- p is the probability of success on a single trial.Since we're dealing with a fair coin, p = 0.5 for heads.But the problem asks for the probability of getting at least 10 heads. That means we need to consider the probabilities of getting exactly 10 heads, exactly 11 heads, and exactly 12 heads. Then, we'll sum those probabilities to get the total probability of getting at least 10 heads.So, let me break it down step by step.1. **Calculate the total number of possible outcomes:** Each flip has 2 possible outcomes, so for 12 flips, the total number of possible outcomes is 2^12. Let me compute that: 2^12 = 4096.2. **Calculate the number of favorable outcomes for each case (10, 11, 12 heads):** - For exactly 10 heads: We need to choose 10 flips out of 12 to be heads. The number of ways to do this is given by the combination C(12, 10). - For exactly 11 heads: Similarly, it's C(12, 11). - For exactly 12 heads: It's C(12, 12). Let me compute each of these combinations. - C(12, 10): I know that C(n, k) = n! / (k!(n - k)!). So, C(12, 10) = 12! / (10! * 2!) = (12 * 11) / (2 * 1) = 66. - C(12, 11): Similarly, C(12, 11) = 12! / (11! * 1!) = 12 / 1 = 12. - C(12, 12): This is 1, since there's only one way to choose all 12 flips to be heads.3. **Sum the number of favorable outcomes:** So, the total number of favorable outcomes is 66 (for 10 heads) + 12 (for 11 heads) + 1 (for 12 heads) = 79.4. **Calculate the probability:** The probability is the number of favorable outcomes divided by the total number of possible outcomes. So, probability = 79 / 4096.Let me double-check my calculations to make sure I didn't make any mistakes.- Total outcomes: 2^12 = 4096. That seems correct.- Combinations: - C(12,10) = 66. Yes, because 12 choose 10 is the same as 12 choose 2, which is (12*11)/2 = 66. - C(12,11) = 12. That makes sense because choosing 11 out of 12 is the same as leaving out 1, and there are 12 ways to leave out 1. - C(12,12) = 1. Correct, as there's only one way to have all 12 heads.- Total favorable: 66 + 12 + 1 = 79. That adds up.- Probability: 79 / 4096. Simplifying that fraction, but since 79 is a prime number, it can't be reduced further.So, the probability is 79/4096. If I wanted to express this as a decimal, I could divide 79 by 4096. Let me do that quickly:79 ÷ 4096 ≈ 0.01923828125.So, approximately a 1.92% chance.Wait, that seems low. Is that correct? Let me think. Getting 10, 11, or 12 heads out of 12 flips is quite rare because it's very close to the maximum number of heads possible. So, a low probability makes sense.Alternatively, I can think about the symmetry of the binomial distribution. Since the coin is fair, the probability of getting at least 10 heads should be the same as the probability of getting at least 10 tails. But since we're dealing with a small number of trials, the probabilities for extreme outcomes are indeed quite low.Another way to verify is to use the binomial probability formula for each case and sum them up.For exactly 10 heads:P(10) = C(12,10) * (0.5)^10 * (0.5)^2 = 66 * (1/1024) * (1/4) = 66 / 4096.Similarly, for exactly 11 heads:P(11) = C(12,11) * (0.5)^11 * (0.5)^1 = 12 * (1/2048) * (1/2) = 12 / 4096.For exactly 12 heads:P(12) = C(12,12) * (0.5)^12 * (0.5)^0 = 1 * (1/4096) * 1 = 1 / 4096.Adding these up:66/4096 + 12/4096 + 1/4096 = (66 + 12 + 1)/4096 = 79/4096.Yes, that matches my earlier calculation. So, I'm confident that the probability is indeed 79/4096.Just to get a sense of the magnitude, 79 divided by 4096 is approximately 0.019238, which is about 1.92%. So, less than 2% chance of getting at least 10 heads in 12 flips. That seems reasonable given how unlikely it is to get so many heads in a row.I think I've covered all the steps and double-checked my work. I don't see any mistakes in my reasoning or calculations.

💡Okay, so I have this problem here: For any real number ( a ), the fixed point through which the line ( y = ax - 3a + 2 ) passes is ______. Hmm, I need to find this fixed point. Let me think about how to approach this.First, I remember that a fixed point means that no matter what value ( a ) takes, the point will always lie on the line. So, if I can find such a point ( (x, y) ) that satisfies the equation for all ( a ), that should be the fixed point.Looking at the equation ( y = ax - 3a + 2 ), I notice that ( a ) is multiplied by both ( x ) and ( -3 ). Maybe I can factor out ( a ) from those terms. Let me try that:( y = a(x - 3) + 2 ).Okay, so now the equation is written as ( y = a(x - 3) + 2 ). This looks like a linear equation where ( a ) is the slope, and ( 2 ) is the y-intercept. But since ( a ) can be any real number, the slope can change, but the line must always pass through a specific point regardless of the slope.To find this fixed point, I need to find values of ( x ) and ( y ) that satisfy the equation for all ( a ). Since ( a ) can be any real number, the coefficient of ( a ) must be zero to ensure that the equation holds true regardless of ( a ). Otherwise, if the coefficient of ( a ) wasn't zero, changing ( a ) would change the value of ( y ), which contradicts the idea of a fixed point.So, let me set the coefficient of ( a ) to zero:( x - 3 = 0 ).Solving this gives:( x = 3 ).Now that I have ( x = 3 ), I can substitute this back into the original equation to find ( y ):( y = a(3) - 3a + 2 ).Simplifying this:( y = 3a - 3a + 2 ).The ( 3a ) and ( -3a ) terms cancel each other out, leaving:( y = 2 ).So, the fixed point is ( (3, 2) ). Let me double-check this to make sure I didn't make a mistake.If I plug ( x = 3 ) into the original equation, regardless of the value of ( a ), the ( a ) terms should cancel out:( y = a(3) - 3a + 2 = 3a - 3a + 2 = 0 + 2 = 2 ).Yep, that works. So, no matter what ( a ) is, when ( x = 3 ), ( y ) will always be 2. That means the point ( (3, 2) ) is indeed the fixed point through which all these lines pass.I think that's the solution. It makes sense because if you imagine changing the slope ( a ), the line will pivot around the fixed point ( (3, 2) ). So, all those lines with different slopes will intersect at that single point.Just to visualize it a bit more, if ( a = 0 ), the equation becomes ( y = 0x - 0 + 2 = 2 ), which is a horizontal line passing through ( (3, 2) ). If ( a = 1 ), the equation is ( y = x - 3 + 2 = x - 1 ), which is a line with a slope of 1 passing through ( (3, 2) ). If ( a = 2 ), it's ( y = 2x - 6 + 2 = 2x - 4 ), which also passes through ( (3, 2) ). So, no matter what ( a ) is, the point ( (3, 2) ) is always on the line.Therefore, I'm confident that the fixed point is ( (3, 2) ).**Final Answer**The fixed point is boxed{(3, 2)}.

💡Okay, so I have this problem here. It says:Given p: |x - a| ≤ 1; q: x² - 5x + 4 ≤ 0. If p is a sufficient condition for q, then the range of values for a is __________.Hmm, let me try to understand what this means. First, I know that in logic, if p is a sufficient condition for q, it means that whenever p is true, q must also be true. In other words, p implies q. So, if |x - a| ≤ 1 is true, then x² - 5x + 4 ≤ 0 must also be true.Alright, so I need to find the values of 'a' such that whenever x is within 1 unit of 'a', x² - 5x + 4 is less than or equal to zero.Let me break this down step by step.First, let's analyze the inequality q: x² - 5x + 4 ≤ 0.I can factor this quadratic equation. Let me see, x² - 5x + 4. The factors of 4 that add up to -5 are -1 and -4. So, it factors to (x - 1)(x - 4) ≤ 0.So, the inequality becomes (x - 1)(x - 4) ≤ 0.To solve this inequality, I can use a sign chart. The critical points are x = 1 and x = 4. These divide the number line into three intervals:1. x < 12. 1 < x < 43. x > 4Now, I'll test each interval:1. For x < 1, say x = 0: (0 - 1)(0 - 4) = (-1)(-4) = 4, which is positive. So, the inequality is not satisfied here.2. For 1 < x < 4, say x = 2: (2 - 1)(2 - 4) = (1)(-2) = -2, which is negative. So, the inequality is satisfied here.3. For x > 4, say x = 5: (5 - 1)(5 - 4) = (4)(1) = 4, which is positive. So, the inequality is not satisfied here.Now, we also need to check the endpoints:At x = 1: (1 - 1)(1 - 4) = 0 * (-3) = 0, which satisfies the inequality.At x = 4: (4 - 1)(4 - 4) = 3 * 0 = 0, which also satisfies the inequality.So, the solution to q is 1 ≤ x ≤ 4. That means x must be between 1 and 4, inclusive.Now, let's look at p: |x - a| ≤ 1.This inequality represents all x such that x is within 1 unit of 'a'. In other words, x lies in the interval [a - 1, a + 1].So, p is true when x is in [a - 1, a + 1].Since p is a sufficient condition for q, it means that if x is in [a - 1, a + 1], then x must also be in [1, 4]. In logical terms, [a - 1, a + 1] must be a subset of [1, 4].Therefore, the interval [a - 1, a + 1] must lie entirely within [1, 4].To ensure this, the left endpoint of [a - 1, a + 1] must be greater than or equal to 1, and the right endpoint must be less than or equal to 4.So, we can write two inequalities:1. a - 1 ≥ 12. a + 1 ≤ 4Let me solve these inequalities one by one.Starting with the first inequality: a - 1 ≥ 1Adding 1 to both sides: a ≥ 2Now, the second inequality: a + 1 ≤ 4Subtracting 1 from both sides: a ≤ 3So, combining both inequalities, we get 2 ≤ a ≤ 3.Therefore, the range of values for 'a' is from 2 to 3, inclusive.Let me double-check to make sure I didn't make any mistakes.If a = 2, then the interval [a - 1, a + 1] becomes [1, 3]. This is entirely within [1, 4], so that works.If a = 3, then the interval [a - 1, a + 1] becomes [2, 4]. This is also entirely within [1, 4], so that works too.What if a is between 2 and 3, say a = 2.5? Then the interval is [1.5, 3.5], which is still within [1, 4]. Perfect.What if a is less than 2, say a = 1? Then the interval is [0, 2]. But 0 is less than 1, which is outside the interval [1, 4]. So, p would not be a sufficient condition for q because some x's in [0, 2] are not in [1, 4].Similarly, if a is greater than 3, say a = 4, the interval becomes [3, 5]. But 5 is greater than 4, so some x's in [3, 5] are not in [1, 4]. Hence, p wouldn't be a sufficient condition for q.Therefore, my conclusion seems solid. The values of 'a' must be between 2 and 3, inclusive.**Final Answer**The range of values for ( a ) is boxed{[2, 3]}.

💡Okay, so I have this problem here about a statement T, which says, "If the sum of the digits of the whole number n is divisible by 9, then n is divisible by 9." And I need to find a value of n from the given options that shows this statement T is false. The options are 54, 81, 90, 117, or none of these.Alright, first, I remember that there's a rule in math that says if the sum of a number's digits is divisible by 9, then the number itself is divisible by 9. So, this statement T is actually a known rule. But the question is asking for a counterexample, which would mean a number where the sum of its digits is divisible by 9, but the number itself is not divisible by 9. If such a number exists among the options, then T is false; otherwise, if none of the options provide such a case, then T holds true, and the answer would be E, none of these.Let me go through each option one by one.Starting with option A: 54. Let's check the sum of its digits. 5 + 4 equals 9. Okay, 9 is divisible by 9, so according to the rule, 54 should be divisible by 9. Let me confirm: 54 divided by 9 is 6, which is an integer. So, 54 is indeed divisible by 9. Therefore, 54 doesn't serve as a counterexample because both the sum and the number itself are divisible by 9.Moving on to option B: 81. The sum of its digits is 8 + 1, which is 9. Again, 9 is divisible by 9, so according to the rule, 81 should be divisible by 9. Let me check: 81 divided by 9 is 9, which is an integer. So, 81 is also divisible by 9. This means 81 doesn't work as a counterexample either.Next is option C: 90. The sum of its digits is 9 + 0, which is 9. Once more, 9 is divisible by 9, so 90 should be divisible by 9. Checking: 90 divided by 9 is 10, which is an integer. So, 90 is divisible by 9 as well. No luck here for a counterexample.Now, option D: 117. Let's add its digits: 1 + 1 + 7 equals 9. Again, 9 is divisible by 9, so according to the rule, 117 should be divisible by 9. Let me divide 117 by 9: 117 divided by 9 is 13, which is an integer. So, 117 is also divisible by 9. This doesn't help us find a counterexample either.So, after checking all the options A through D, each time the sum of the digits is 9, which is divisible by 9, and the number itself is also divisible by 9. Therefore, none of these options provide a counterexample to the statement T. Wait, but the question is asking for a value of n that shows T to be false. Since all the options provided do not contradict the statement T, the correct answer must be E, none of these.Just to make sure I'm not missing something, let me think if there's any number where the sum of its digits is divisible by 9, but the number itself isn't. For example, take the number 18: sum is 9, and 18 is divisible by 9. How about 27? Sum is 9, and 27 is divisible by 9. Hmm, seems like this rule holds true for these numbers. Maybe it's a general rule after all.I think I've heard before that this is a divisibility rule for 9, so it's not just a coincidence with these numbers. Therefore, it's unlikely that any number with a digit sum divisible by 9 isn't itself divisible by 9. So, in this case, none of the given options can disprove the statement T.So, I'm confident that the answer is E, none of these.

💡First, I need to find the value of ( x ) from the equation ( 5^x = 3 ). By taking the logarithm base 5 of both sides, I can express ( x ) as ( x = log_{5}3 ).Next, I need to determine the value of ( y ), which is given by ( y = log_{5}frac{9}{25} ). I can simplify this logarithm by recognizing that ( frac{9}{25} ) is ( frac{3^2}{5^2} ), so ( y = log_{5}frac{3^2}{5^2} ).Using the logarithmic identity ( log_{a}frac{b}{c} = log_{a}b - log_{a}c ), I can rewrite ( y ) as ( y = log_{5}3^2 - log_{5}5^2 ). Simplifying further, this becomes ( y = 2log_{5}3 - 2 ).Now, I need to calculate ( 2x - y ). Substituting the expressions for ( x ) and ( y ), I get:[2x - y = 2log_{5}3 - (2log_{5}3 - 2) = 2]Thus, the value of ( 2x - y ) is 2.

💡Alright, so I have this geometry problem here about triangle PQR. It's a right-angled triangle at R, and the length of PQ is 10 units. They've constructed squares PQXY and PRWZ outside the triangle. The key point is that the points X, Y, Z, and W lie on a circle. I need to find the perimeter of triangle PQR.First, let me try to visualize the problem. Triangle PQR is right-angled at R, so sides PR and RQ are the legs, and PQ is the hypotenuse, which is given as 10 units. Squares are constructed on sides PQ and PR, outside the triangle. So, square PQXY is built on PQ, and square PRWZ is built on PR.Now, the points X, Y, Z, and W lie on a circle. That means these four points are concyclic or lie on the same circumference. This is an important clue because it tells me that there must be some relationship between these points that allows them to lie on a circle. Maybe some properties of cyclic quadrilaterals can be applied here.Let me recall that in a cyclic quadrilateral, the opposite angles sum to 180 degrees. But since these points are vertices of squares, which have right angles, perhaps there's something about the angles at these points that can help.But before diving into that, maybe it's better to assign some coordinates to the triangle to make things more concrete. Let me place point R at the origin (0,0) for simplicity. Since it's a right-angled triangle at R, let me assume point P is on the x-axis and point Q is on the y-axis. So, point P would be (a, 0) and point Q would be (0, b), where a and b are the lengths of sides PR and RQ, respectively.Given that PQ is the hypotenuse and is 10 units, by the Pythagorean theorem, we have:a² + b² = 10² = 100.So, the perimeter of triangle PQR would be a + b + 10. Therefore, if I can find the values of a and b, I can compute the perimeter.Now, let's think about the squares constructed on PQ and PR. Square PQXY is constructed outside the triangle on PQ. Since PQ is the hypotenuse, which is 10 units, each side of square PQXY is also 10 units. Similarly, square PRWZ is constructed on PR, which is of length a, so each side of square PRWZ is a units.Let me try to figure out the coordinates of points X, Y, Z, and W.Starting with square PQXY. Since PQ is from point P(a, 0) to Q(0, b), the square is constructed outside the triangle. To find the coordinates of X and Y, I need to figure out how the square is oriented.One way to do this is to consider the direction of the square. Since it's constructed outside the triangle, the square PQXY will be on the side opposite to the triangle. So, from point P(a, 0), moving in a direction perpendicular to PQ to construct the square.Similarly, for square PRWZ, constructed on PR. Since PR is along the x-axis from R(0,0) to P(a, 0), the square PRWZ will be constructed above or below this side. Since it's constructed outside the triangle, and the triangle is above the x-axis (since Q is at (0, b)), the square PRWZ should be constructed below the x-axis.Wait, actually, the triangle is right-angled at R, so if P is at (a, 0) and Q is at (0, b), then the triangle is in the first quadrant. So, constructing squares outside the triangle would mean that square PQXY is constructed in such a way that it doesn't overlap with the triangle, and similarly for square PRWZ.This is getting a bit abstract. Maybe assigning specific coordinates will help.Let me denote:- Point R: (0, 0)- Point P: (a, 0)- Point Q: (0, b)Now, square PQXY is constructed outside the triangle on PQ. To find the coordinates of X and Y, I need to figure out the direction in which the square extends from PQ.The vector from P to Q is (-a, b). To construct a square, we need to find points X and Y such that PQXY is a square. To do this, we can rotate the vector PQ by 90 degrees to get the direction of the next side.Rotating vector PQ = (-a, b) by 90 degrees counterclockwise would give (-b, -a). So, starting from Q(0, b), moving in the direction of (-b, -a) gives point X. Similarly, starting from P(a, 0), moving in the direction of (-b, -a) gives point Y.Wait, actually, rotating PQ by 90 degrees counterclockwise would give a vector perpendicular to PQ. But since the square is constructed outside the triangle, we need to make sure the rotation is in the correct direction.Alternatively, perhaps it's easier to use complex numbers to find the coordinates.Let me represent points as complex numbers. Let me denote point P as a + 0i and point Q as 0 + bi.The vector PQ is Q - P = (-a + bi). To construct square PQXY, we need to find points X and Y such that PQXY is a square. To do this, we can rotate the vector PQ by 90 degrees to get the direction of the next side.Rotating (-a + bi) by 90 degrees counterclockwise is equivalent to multiplying by i, which gives (-a + bi) * i = -ai - b. So, the vector from Q to X is (-ai - b). Therefore, point X is Q + (-ai - b) = (0 + bi) + (-ai - b) = (-b - ai).Similarly, the vector from P to Y is the same as the vector from Q to X, which is (-ai - b). So, point Y is P + (-ai - b) = (a + 0i) + (-ai - b) = (a - b) - ai.Wait, but these are complex numbers, so to get the coordinates, I need to separate the real and imaginary parts.Point X: (-b, -a)Point Y: (a - b, -a)Similarly, for square PRWZ constructed on PR. Vector PR is P - R = (a, 0). Rotating this vector by 90 degrees counterclockwise would give (0, a). So, starting from R(0,0), moving in the direction of (0, a) gives point W. But since the square is constructed outside the triangle, which is in the first quadrant, the square PRWZ should be constructed below the x-axis.Wait, if we rotate PR by 90 degrees clockwise, we get (0, -a). So, starting from R(0,0), moving in the direction of (0, -a) gives point W(0, -a). Then, from P(a, 0), moving in the same direction gives point Z(a, -a).Wait, let me verify that.Vector PR is (a, 0). Rotating it 90 degrees clockwise gives (0, -a). So, starting from R(0,0), moving in the direction of (0, -a) gives point W(0, -a). Then, from P(a, 0), moving in the same direction gives point Z(a, -a).So, square PRWZ has points P(a, 0), R(0,0), W(0, -a), and Z(a, -a).Now, we have the coordinates of points X, Y, Z, and W:- X: (-b, -a)- Y: (a - b, -a)- Z: (a, -a)- W: (0, -a)Wait, let me double-check the coordinates of Y. Earlier, I thought it was (a - b, -a), but let me verify.From point P(a, 0), moving in the direction of (-ai - b) in complex plane terms translates to subtracting b from the real part and subtracting a from the imaginary part, so (a - b, 0 - a) = (a - b, -a). Yes, that seems correct.So, points X(-b, -a), Y(a - b, -a), Z(a, -a), and W(0, -a) lie on a circle.Now, since all four points lie on a circle, the circle must pass through these four points. Let's see if we can find the equation of the circle passing through these points.First, let's note that points W, Z, and Y all have the same y-coordinate, which is -a. So, points W(0, -a), Z(a, -a), and Y(a - b, -a) lie on the horizontal line y = -a. However, point X(-b, -a) also lies on this line. Wait, that can't be right because if all four points lie on the same horizontal line, they can't lie on a circle unless the circle is degenerate, which it's not because a circle is determined by three non-collinear points.Wait, this suggests that my earlier assumption about the coordinates might be incorrect because if all four points lie on the same horizontal line, they can't lie on a circle unless it's a straight line, which contradicts the problem statement.Hmm, so perhaps my coordinate assignments are wrong. Maybe I made a mistake in determining the coordinates of points X, Y, Z, and W.Let me try a different approach. Instead of assigning coordinates immediately, maybe I can use properties of squares and cyclic quadrilaterals.Since squares are constructed on PQ and PR, and points X, Y, Z, W lie on a circle, perhaps there is some symmetry or specific angle relationships that can be used.In a square, all sides are equal and all angles are 90 degrees. So, the sides of the squares are equal to the sides of the triangle they are constructed upon.Given that, perhaps the distances from the center of the circle to each of these points are equal, which is the definition of a circle.Alternatively, maybe the circle is the circumcircle of the quadrilateral XY WZ, and since it's a square, perhaps the quadrilateral is cyclic, but squares are always cyclic because all their vertices lie on a circle (they are rectangles, which are cyclic).Wait, but in this case, the quadrilateral XY WZ is not a square, but a combination of two squares. So, it's a four-sided figure with sides from two different squares.Wait, maybe I can consider the fact that the diagonals of the squares intersect at the center of the circle.In square PQXY, the diagonals intersect at the midpoint of PQ, which would be the center of the square. Similarly, in square PRWZ, the diagonals intersect at the midpoint of PR, which would be the center of that square.But since points X, Y, Z, W lie on a single circle, perhaps the centers of these squares coincide at the center of the circle.Wait, that might not necessarily be true because the squares are constructed on different sides.Alternatively, maybe the center of the circle is the midpoint of PQ, which is also the midpoint of the hypotenuse of the right-angled triangle. In a right-angled triangle, the midpoint of the hypotenuse is equidistant from all three vertices. So, perhaps this midpoint is the center of the circle passing through X, Y, Z, W.Let me explore this idea.In triangle PQR, right-angled at R, the midpoint of hypotenuse PQ is equidistant from P, Q, and R. Let's denote this midpoint as O. So, O is the center of the circle passing through P, Q, and R.But in this problem, the circle passes through X, Y, Z, and W, not P, Q, R. So, perhaps O is also the center of the circle passing through X, Y, Z, and W.If that's the case, then the distances from O to each of X, Y, Z, and W must be equal.Let me try to compute the distances from O to each of these points.First, let's find the coordinates of O, the midpoint of PQ.Since P is (a, 0) and Q is (0, b), the midpoint O is ((a + 0)/2, (0 + b)/2) = (a/2, b/2).Now, let's find the coordinates of X, Y, Z, and W again, but this time, perhaps more carefully.Starting with square PQXY. Since PQ is from P(a, 0) to Q(0, b), the square is constructed outside the triangle. To find points X and Y, we need to construct the square such that PQ is one side, and X and Y are the other two vertices.One way to find X and Y is to rotate the vector PQ by 90 degrees to get the direction of the next side.The vector PQ is Q - P = (-a, b). Rotating this vector 90 degrees counterclockwise gives (-b, -a). So, starting from Q(0, b), moving in the direction of (-b, -a) gives point X.So, point X = Q + (-b, -a) = (0 - b, b - a) = (-b, b - a).Similarly, starting from P(a, 0), moving in the direction of (-b, -a) gives point Y.So, point Y = P + (-b, -a) = (a - b, 0 - a) = (a - b, -a).Now, for square PRWZ constructed on PR. Vector PR is R - P = (-a, 0). Rotating this vector 90 degrees counterclockwise gives (0, a). So, starting from R(0, 0), moving in the direction of (0, a) gives point W.Wait, but since the square is constructed outside the triangle, which is in the first quadrant, moving in the positive y-direction from R would still be in the first quadrant, but the square should be constructed outside the triangle, which is already in the first quadrant. Maybe it's constructed in the negative y-direction.Alternatively, rotating vector PR 90 degrees clockwise gives (0, -a). So, starting from R(0, 0), moving in the direction of (0, -a) gives point W(0, -a). Then, from P(a, 0), moving in the same direction gives point Z(a, -a).So, square PRWZ has points P(a, 0), R(0, 0), W(0, -a), and Z(a, -a).Now, we have the coordinates of all four points:- X(-b, b - a)- Y(a - b, -a)- Z(a, -a)- W(0, -a)Now, we need to find the distances from the center O(a/2, b/2) to each of these points and set them equal since they lie on the same circle.Let's compute the distance from O to X, O to Y, O to Z, and O to W.Distance from O to X:OX² = (-b - a/2)² + (b - a - b/2)²= (-b - a/2)² + (-a - b/2)²= (b + a/2)² + (a + b/2)²= (a²/4 + ab + b²) + (a² + ab + b²/4)= a²/4 + ab + b² + a² + ab + b²/4= (a²/4 + a²) + (ab + ab) + (b² + b²/4)= (5a²/4) + 2ab + (5b²/4)Distance from O to Y:OY² = (a - b - a/2)² + (-a - b/2)²= (a/2 - b)² + (-a - b/2)²= (a²/4 - ab + b²) + (a² + ab + b²/4)= a²/4 - ab + b² + a² + ab + b²/4= (a²/4 + a²) + (-ab + ab) + (b² + b²/4)= (5a²/4) + 0 + (5b²/4)Distance from O to Z:OZ² = (a - a/2)² + (-a - b/2)²= (a/2)² + (-a - b/2)²= a²/4 + (a + b/2)²= a²/4 + a² + ab + b²/4= (a²/4 + a²) + ab + b²/4= (5a²/4) + ab + (b²/4)Distance from O to W:OW² = (0 - a/2)² + (-a - b/2)²= (a²/4) + (-a - b/2)²= a²/4 + (a + b/2)²= a²/4 + a² + ab + b²/4= (a²/4 + a²) + ab + b²/4= (5a²/4) + ab + (b²/4)Wait a minute, so OX² = (5a²/4) + 2ab + (5b²/4)OY² = (5a²/4) + (5b²/4)OZ² = (5a²/4) + ab + (b²/4)OW² = (5a²/4) + ab + (b²/4)Since all four points lie on the same circle, their distances from the center O must be equal. Therefore, OX² = OY² = OZ² = OW².Looking at OX² and OY²:OX² = (5a²/4) + 2ab + (5b²/4)OY² = (5a²/4) + (5b²/4)Setting OX² = OY²:(5a²/4) + 2ab + (5b²/4) = (5a²/4) + (5b²/4)Subtracting (5a²/4) + (5b²/4) from both sides:2ab = 0But a and b are lengths of sides of a triangle, so they can't be zero. Therefore, this leads to a contradiction.Hmm, that suggests that my earlier assumption about the coordinates might still be incorrect. Maybe I made a mistake in determining the direction of the squares.Perhaps instead of rotating the vectors in a particular direction, I should consider the other direction.Let me try rotating the vector PQ by 90 degrees clockwise instead of counterclockwise.Vector PQ is (-a, b). Rotating this 90 degrees clockwise gives (b, a). So, starting from Q(0, b), moving in the direction of (b, a) gives point X.So, point X = Q + (b, a) = (0 + b, b + a) = (b, a + b)Similarly, starting from P(a, 0), moving in the direction of (b, a) gives point Y.So, point Y = P + (b, a) = (a + b, 0 + a) = (a + b, a)Now, for square PRWZ, let's rotate vector PR 90 degrees clockwise. Vector PR is (-a, 0). Rotating this 90 degrees clockwise gives (0, -a). So, starting from R(0, 0), moving in the direction of (0, -a) gives point W(0, -a). Then, from P(a, 0), moving in the same direction gives point Z(a, -a).So, square PRWZ has points P(a, 0), R(0, 0), W(0, -a), and Z(a, -a).Now, the coordinates of points X, Y, Z, W are:- X(b, a + b)- Y(a + b, a)- Z(a, -a)- W(0, -a)Now, let's compute the distances from O(a/2, b/2) to each of these points.Distance from O to X:OX² = (b - a/2)² + (a + b - b/2)²= (b - a/2)² + (a + b/2)²= (b² - ab + a²/4) + (a² + ab + b²/4)= b² - ab + a²/4 + a² + ab + b²/4= (a²/4 + a²) + (-ab + ab) + (b² + b²/4)= (5a²/4) + 0 + (5b²/4)Distance from O to Y:OY² = (a + b - a/2)² + (a - b/2)²= (a/2 + b)² + (a - b/2)²= (a²/4 + ab + b²) + (a² - ab + b²/4)= a²/4 + ab + b² + a² - ab + b²/4= (a²/4 + a²) + (ab - ab) + (b² + b²/4)= (5a²/4) + 0 + (5b²/4)Distance from O to Z:OZ² = (a - a/2)² + (-a - b/2)²= (a/2)² + (-a - b/2)²= a²/4 + (a + b/2)²= a²/4 + a² + ab + b²/4= (a²/4 + a²) + ab + b²/4= (5a²/4) + ab + (b²/4)Distance from O to W:OW² = (0 - a/2)² + (-a - b/2)²= (a²/4) + (-a - b/2)²= a²/4 + (a + b/2)²= a²/4 + a² + ab + b²/4= (a²/4 + a²) + ab + b²/4= (5a²/4) + ab + (b²/4)Now, we have:OX² = OY² = (5a²/4) + (5b²/4)OZ² = OW² = (5a²/4) + ab + (b²/4)Since all four points lie on the same circle, OX² = OY² = OZ² = OW².Therefore, we have:(5a²/4) + (5b²/4) = (5a²/4) + ab + (b²/4)Subtracting (5a²/4) from both sides:(5b²/4) = ab + (b²/4)Subtracting (b²/4) from both sides:(5b²/4 - b²/4) = ab(4b²/4) = abb² = abDividing both sides by b (since b ≠ 0):b = aSo, we have b = a.Given that, from the Pythagorean theorem:a² + b² = 100a² + a² = 1002a² = 100a² = 50a = √50 = 5√2Therefore, a = b = 5√2.Now, the perimeter of triangle PQR is:PQ + PR + RQ = 10 + a + b = 10 + 5√2 + 5√2 = 10 + 10√2So, the perimeter is 10 + 10√2, which corresponds to option C.