💡Okay, so I have this problem about a right triangle PQR with sides PQ = 20, PR = 21, and QR = 29. I need to find the length of PI, where I is the incenter of the triangle. Hmm, the incenter is the point where the angle bisectors meet, right? So, PI is the distance from vertex P to the incenter.First, I remember that in a right triangle, the inradius can be found using a specific formula. Let me recall... I think it's related to the legs and the hypotenuse. Maybe something like r = (a + b - c)/2, where a and b are the legs and c is the hypotenuse. Let me check if that makes sense.Given that PQ = 20 and PR = 21 are the legs, and QR = 29 is the hypotenuse. Plugging into the formula: r = (20 + 21 - 29)/2. Let me calculate that: 20 + 21 is 41, minus 29 is 12, divided by 2 is 6. So, the inradius r is 6. Wait, but is the inradius the same as PI? I think the inradius is the distance from the incenter to any side of the triangle, but PI is the distance from vertex P to the incenter. So, maybe I need a different approach.I remember there's a formula for the distance from a vertex to the incenter. It involves the sides of the triangle and the angles. But since this is a right triangle, maybe I can use some properties specific to right triangles.Let me think. In a right triangle, the inradius is r = (a + b - c)/2, which we already found to be 6. Now, to find PI, which is the distance from P to I. Maybe I can use coordinates to solve this.Let me place the triangle on a coordinate system. Let’s put point P at the origin (0,0), Q at (20,0), and R at (0,21). Then, the incenter I can be found using the formula for coordinates of the incenter: ( (aA + bB + cC)/ (a + b + c) ), where a, b, c are the lengths of the sides opposite to vertices A, B, C respectively.Wait, in triangle PQR, sides opposite to P, Q, R are QR, PR, PQ respectively. So, QR is 29, PR is 21, PQ is 20. So, the coordinates of I would be ( (29*0 + 21*20 + 20*0)/ (29 + 21 + 20), (29*0 + 21*0 + 20*21)/ (29 + 21 + 20) ). Let me compute that.First, the denominator is 29 + 21 + 20 = 70. The x-coordinate is (29*0 + 21*20 + 20*0)/70 = (0 + 420 + 0)/70 = 420/70 = 6. The y-coordinate is (29*0 + 21*0 + 20*21)/70 = (0 + 0 + 420)/70 = 6. So, the incenter I is at (6,6).Now, point P is at (0,0), so the distance PI is the distance from (0,0) to (6,6). Using the distance formula: sqrt[(6-0)^2 + (6-0)^2] = sqrt[36 + 36] = sqrt[72] = 6*sqrt(2). Wait, that's approximately 8.485, but the options are 6,7,8,9. Hmm, that doesn't match. Did I make a mistake?Wait, maybe I confused the formula for the inradius with the coordinates. Let me double-check. The inradius formula r = (a + b - c)/2 is correct for a right triangle, giving r = 6. But when I calculated the coordinates, I got I at (6,6), which is 6*sqrt(2) away from P. But 6*sqrt(2) is about 8.485, which isn't one of the options. The closest option is 6, but that's the inradius, not PI.Maybe I should use another formula for the distance from the vertex to the incenter. I think the formula is PI = sqrt(r^2 + (s - p)^2), where s is the semiperimeter and p is the side opposite to the vertex. Wait, let me recall the exact formula.Alternatively, I remember that in any triangle, the distance from a vertex to the incenter can be found using the formula: d = sqrt(r^2 + (s - a)^2), where s is the semiperimeter and a is the side opposite the vertex. Let me try that.First, the semiperimeter s = (a + b + c)/2 = (20 + 21 + 29)/2 = 70/2 = 35. The side opposite vertex P is QR = 29. So, s - a = 35 - 29 = 6. The inradius r is 6. So, PI = sqrt(6^2 + 6^2) = sqrt(36 + 36) = sqrt(72) = 6*sqrt(2), which is the same as before. Hmm, still not matching the options.Wait, maybe I'm overcomplicating this. Since the triangle is right-angled at P, maybe there's a simpler way. In a right triangle, the inradius is r = (a + b - c)/2, which is 6. The incenter is located at a distance r from each side. So, from P, which is the right angle, the incenter is r units away along both legs. So, the coordinates would be (r, r) = (6,6), as I found earlier.But the distance from P(0,0) to I(6,6) is sqrt(6^2 + 6^2) = 6*sqrt(2), which is approximately 8.485. But the options are 6,7,8,9. Since 6*sqrt(2) is about 8.485, which is closest to 8. But 8 isn't the inradius. Wait, maybe the question is asking for the inradius, but it's specified as PI, which is the distance from P to I.Alternatively, maybe I made a mistake in the coordinate system. Let me try a different approach. Maybe using the formula for the length of the angle bisector.The formula for the length of the angle bisector from vertex P is given by:PI = (2ab cos(theta/2)) / (a + b)Where a and b are the sides adjacent to angle P, and theta is the angle at P. Since it's a right angle, theta = 90 degrees, so theta/2 = 45 degrees. Cos(45) is sqrt(2)/2.So, PI = (2 * 20 * 21 * sqrt(2)/2) / (20 + 21) = (420 * sqrt(2)/2) / 41 = (210 * sqrt(2)) / 41 ≈ (210 * 1.4142)/41 ≈ 296.982/41 ≈ 7.243. Hmm, that's approximately 7.24, which is close to 7. But the options are 6,7,8,9.Wait, but earlier I got 6*sqrt(2) ≈ 8.485, and now I'm getting ≈7.24. These are two different results. I must be confusing different formulas.Let me check the angle bisector formula again. The formula for the length of the angle bisector from angle P is:PI = (2ab / (a + b)) * cos(theta/2)Which is what I used. Since theta is 90 degrees, cos(theta/2) is sqrt(2)/2. So, PI = (2*20*21)/(20+21) * sqrt(2)/2 = (840/41) * sqrt(2)/2 = (420/41) * sqrt(2) ≈ (10.2439) * 1.4142 ≈ 14.5. Wait, that can't be right because the triangle's sides are 20,21,29, so the inradius is 6, and the incenter is inside the triangle, so PI should be less than 20 or 21.Wait, I think I messed up the formula. Let me look up the correct formula for the length of the angle bisector in a triangle.The correct formula for the length of the angle bisector from angle P is:PI = (2ab / (a + b)) * cos(theta/2)But in this case, theta is 90 degrees, so cos(theta/2) is sqrt(2)/2. So, PI = (2*20*21)/(20+21) * sqrt(2)/2 = (840/41) * sqrt(2)/2 = (420/41) * sqrt(2) ≈ (10.2439) * 1.4142 ≈ 14.5. That still doesn't make sense because the inradius is 6, and the incenter is inside the triangle, so PI should be less than 20 or 21.Wait, maybe I'm confusing the angle bisector with the inradius. The inradius is the distance from the incenter to the sides, but PI is the distance from P to I, which is different.Alternatively, maybe I should use coordinates again. If I is at (6,6), then PI is sqrt(6^2 + 6^2) = 6*sqrt(2) ≈ 8.485. Since 8.485 is approximately 8.5, which is closest to 8, but the options are 6,7,8,9. So, maybe the answer is 8. But earlier, using the angle bisector formula, I got approximately 7.24, which is closer to 7.Wait, perhaps I should use the formula for the distance from the vertex to the incenter in terms of the sides and angles. The formula is:PI = (r) / sin(theta/2)Where r is the inradius, and theta is the angle at P. Since theta is 90 degrees, sin(theta/2) is sin(45) = sqrt(2)/2. So, PI = 6 / (sqrt(2)/2) = 6 * 2 / sqrt(2) = 12 / sqrt(2) = 6*sqrt(2) ≈ 8.485. So, again, approximately 8.485, which is about 8.5.But the options are 6,7,8,9. So, 8 is the closest. But wait, the inradius is 6, which is the distance from I to the sides, not from P to I. So, maybe the answer is 6, but that's the inradius, not PI.Wait, I'm getting confused. Let me clarify:- Inradius (r) = 6, which is the distance from I to each side.- PI is the distance from P to I, which is sqrt(r^2 + r^2) = r*sqrt(2) = 6*sqrt(2) ≈ 8.485.But 8.485 isn't an option. The closest option is 8, but the exact value is 6*sqrt(2). Wait, 6*sqrt(2) is approximately 8.485, which is more than 8 but less than 9. Since 8.485 is closer to 8.5, which is between 8 and 9, but the options are integers. Maybe the answer is 6, but that's the inradius, not PI.Wait, maybe I made a mistake in the coordinate system. Let me try again. If P is at (0,0), Q at (20,0), R at (0,21), then the incenter I is at (r, r) = (6,6). So, the distance from P(0,0) to I(6,6) is sqrt(6^2 + 6^2) = sqrt(72) = 6*sqrt(2). So, that's correct.But the options are 6,7,8,9. So, 6*sqrt(2) is approximately 8.485, which is closest to 8. But 8 is an option, so maybe the answer is 8. Alternatively, perhaps the question is asking for the inradius, which is 6, but it's specified as PI, which is the distance from P to I.Wait, maybe I should use the formula for the distance from the vertex to the incenter in a right triangle. I think it's given by:PI = (a + b - c)/2 * sqrt(2)Where a and b are the legs, and c is the hypotenuse. So, (20 + 21 - 29)/2 = 12/2 = 6. Then, 6*sqrt(2) ≈ 8.485. So, again, the same result.But since 8.485 isn't an option, maybe the answer is 6, but that's the inradius. Alternatively, perhaps I should consider that in a right triangle, the distance from the right angle to the incenter is r*sqrt(2), which is 6*sqrt(2). But since that's not an option, maybe the answer is 6.Wait, but the question specifically asks for PI, the distance from P to I, which is 6*sqrt(2). Since 6*sqrt(2) is approximately 8.485, and the options are 6,7,8,9, the closest integer is 8. So, maybe the answer is 8.Alternatively, perhaps I should use the formula for the length of the angle bisector in a right triangle. The formula is:PI = (2ab)/(a + b) * cos(theta/2)Where theta is 90 degrees, so cos(theta/2) = sqrt(2)/2. So,PI = (2*20*21)/(20+21) * sqrt(2)/2 = (840/41) * sqrt(2)/2 = (420/41) * sqrt(2) ≈ (10.2439) * 1.4142 ≈ 14.5. Wait, that can't be right because the triangle's sides are 20,21,29, so the inradius is 6, and the incenter is inside the triangle, so PI should be less than 20 or 21.Wait, I think I messed up the formula. Let me look up the correct formula for the length of the angle bisector in a right triangle.Actually, in a right triangle, the length of the angle bisector from the right angle can be calculated using the formula:PI = (a + b - c)/2 * sqrt(2)Which is the same as r*sqrt(2), since r = (a + b - c)/2. So, PI = r*sqrt(2) = 6*sqrt(2) ≈ 8.485.Since the options are integers, and 8.485 is closest to 8, I think the answer is 8. But I'm not entirely sure because the exact value is 6*sqrt(2), which isn't an integer. Maybe the question expects the exact value, but the options are integers, so perhaps I made a mistake somewhere.Wait, let me think differently. Maybe I can use the formula for the distance from the vertex to the incenter in terms of the sides and the semiperimeter.The formula is:PI = sqrt(r^2 + (s - a)^2)Where r is the inradius, s is the semiperimeter, and a is the side opposite the vertex.In this case, s = 35, a = QR = 29, so s - a = 6. r = 6. So,PI = sqrt(6^2 + 6^2) = sqrt(72) = 6*sqrt(2) ≈ 8.485.Again, same result. So, I think the answer should be 6*sqrt(2), but since that's not an option, and the closest integer is 8, I'll go with 8.But wait, the options are A.6, B.7, C.8, D.9. So, 8 is an option. Therefore, the answer is 8.

💡Okay, so I have this problem where I'm given that tan x plus tan y equals 15, and cot x plus cot y equals 20. I need to find tan(x + y). Hmm, let's see. I remember that there are some trigonometric identities that relate tan and cot, and also formulas for tan(x + y). Maybe I can use those to solve this.First, let me write down what I know. The problem says:tan x + tan y = 15andcot x + cot y = 20.I need to find tan(x + y). I remember that the formula for tan(x + y) is:tan(x + y) = (tan x + tan y) / (1 - tan x tan y).So, if I can find tan x tan y, then I can plug in the values I have into this formula.Looking at the second equation, cot x + cot y = 20. I know that cot is the reciprocal of tan, so cot x = 1/tan x and cot y = 1/tan y. So, I can rewrite the second equation as:1/tan x + 1/tan y = 20.Hmm, maybe I can combine these fractions. Let's see:( tan y + tan x ) / (tan x tan y) = 20.Oh, wait! The numerator here is tan x + tan y, which is given as 15. So, substituting that in, I get:15 / (tan x tan y) = 20.So, if I solve for tan x tan y, I can multiply both sides by tan x tan y and then divide both sides by 20:15 = 20 * tan x tan ySo, tan x tan y = 15 / 20 = 3/4.Alright, so now I have tan x + tan y = 15 and tan x tan y = 3/4. Perfect, because the formula for tan(x + y) requires both of these.Plugging these into the formula:tan(x + y) = (15) / (1 - 3/4).Let me compute the denominator first: 1 - 3/4 is 1/4.So, tan(x + y) = 15 / (1/4) = 15 * 4 = 60.Wait, that seems straightforward. Let me double-check my steps to make sure I didn't make a mistake.1. I started with tan x + tan y = 15 and cot x + cot y = 20.2. I converted cot x + cot y into (tan x + tan y)/(tan x tan y) = 20.3. Substituted tan x + tan y = 15 into that equation to get 15/(tan x tan y) = 20.4. Solved for tan x tan y and got 3/4.5. Plugged tan x + tan y and tan x tan y into the tan(x + y) formula.6. Calculated the denominator as 1 - 3/4 = 1/4.7. Divided 15 by 1/4, which is the same as multiplying by 4, resulting in 60.Everything seems to check out. I don't see any errors in the calculations or the logic. So, I think tan(x + y) is indeed 60.Just to make sure, let me think if there's another way to approach this problem. Maybe using sine and cosine? Let's see.I know that tan x = sin x / cos x and cot x = cos x / sin x. So, maybe I can express everything in terms of sine and cosine.Given tan x + tan y = 15, which is (sin x / cos x) + (sin y / cos y) = 15.Similarly, cot x + cot y = 20 is (cos x / sin x) + (cos y / sin y) = 20.Hmm, this seems more complicated. Maybe I can find a relationship between sin x, sin y, cos x, and cos y.Alternatively, perhaps I can use the identity for tan(x + y) directly. Wait, I already did that. Maybe another identity? Not sure.Alternatively, let me consider using substitution. Let me let a = tan x and b = tan y. Then, the given equations become:a + b = 15and1/a + 1/b = 20.Which is the same as (a + b)/(ab) = 20.So, substituting a + b = 15, we get 15/(ab) = 20, so ab = 15/20 = 3/4, which is the same as before.Then, tan(x + y) = (a + b)/(1 - ab) = 15/(1 - 3/4) = 15/(1/4) = 60.So, same result. So, that confirms it again.I think I'm confident that the answer is 60. I don't see any alternative interpretations or mistakes in my reasoning.**Final Answer**The value of tan(x+y) is boxed{60}.

💡Okay, so I have this problem here: If x and y are real numbers, and y is equal to the square root of (x minus 3) plus the square root of (3 minus x) plus 8, then I need to find the cube root of (x plus 3y). Hmm, that sounds a bit tricky, but let me try to break it down step by step.First off, I know that square roots can only take non-negative values. So, for the square roots in the equation to be real numbers, the expressions inside them must be greater than or equal to zero. That means for the first square root, sqrt(x - 3), the expression inside, which is (x - 3), must be greater than or equal to zero. So, x - 3 ≥ 0, which simplifies to x ≥ 3. Got that.Now, looking at the second square root, sqrt(3 - x), the expression inside here is (3 - x). For this to be non-negative, we need 3 - x ≥ 0. That simplifies to x ≤ 3. Okay, so from the first square root, x has to be at least 3, and from the second square root, x has to be at most 3. So, putting those two together, x must be exactly 3. That makes sense because it's the only number that's both greater than or equal to 3 and less than or equal to 3.Alright, so x is 3. Now, let's plug that back into the equation for y. So, y equals sqrt(3 - 3) plus sqrt(3 - 3) plus 8. Simplifying that, sqrt(0) is 0, and sqrt(0) is also 0. So, y is 0 + 0 + 8, which is 8. Easy enough.Now, the problem asks for the cube root of (x + 3y). Let's compute x + 3y first. Since x is 3 and y is 8, that becomes 3 + 3*8. Calculating that, 3*8 is 24, so 3 + 24 is 27. So, x + 3y is 27.Finally, we need the cube root of 27. I remember that the cube root of a number is a value that, when multiplied by itself three times, gives the original number. So, what number multiplied by itself three times equals 27? Well, 3 times 3 is 9, and 9 times 3 is 27. So, the cube root of 27 is 3.Wait, let me double-check that to make sure I didn't make any mistakes. So, x had to be 3 because of the domain restrictions on the square roots. Plugging x = 3 into y gives y = 8. Then, x + 3y is 3 + 24, which is 27. The cube root of 27 is indeed 3 because 3 cubed is 27. Yeah, that all checks out.I think I've got it. The cube root of (x + 3y) is 3.

💡First, I recognize that the shirt was sold at a 25% discount, meaning Smith paid 75% of the original price.I'll let ( P ) represent the original selling price of the shirt.Since 75% of ( P ) equals Rs. 560, I can set up the equation:[ 0.75 times P = 560 ]To find ( P ), I'll divide both sides of the equation by 0.75:[ P = frac{560}{0.75} ]Calculating the division:[ P = 746.67 ]Therefore, the original selling price of the shirt before the discount was Rs. 746.67.

💡Okay, so I have this problem where I need to consider the line y = x + b and see where it intersects the ellipse given by (x²)/2 + y² = 1. There are two parts: first, finding the range of values for b such that the line intersects the ellipse at two distinct points, and second, calculating the length of the segment AB when b is 1.Starting with part (1). I remember that to find the points of intersection between a line and an ellipse, I can substitute the equation of the line into the ellipse equation. That should give me a quadratic equation in terms of x, and then I can analyze the discriminant to ensure there are two distinct real solutions.So, substituting y = x + b into the ellipse equation:(x²)/2 + (x + b)² = 1.Let me expand that:(x²)/2 + (x² + 2bx + b²) = 1.Combine like terms:(x²)/2 + x² + 2bx + b² = 1.First, let me convert (x²)/2 to (1/2)x² and x² is just (2/2)x², so adding them together gives (3/2)x². So:(3/2)x² + 2bx + b² - 1 = 0.To make it easier, I can multiply the entire equation by 2 to eliminate the fraction:3x² + 4bx + 2b² - 2 = 0.So now I have a quadratic equation in x: 3x² + 4bx + (2b² - 2) = 0.For this quadratic to have two distinct real solutions, the discriminant must be positive. The discriminant Δ of a quadratic ax² + bx + c is given by Δ = b² - 4ac.In this case, a = 3, b = 4b, and c = 2b² - 2.So, Δ = (4b)² - 4*3*(2b² - 2).Calculating that:Δ = 16b² - 12*(2b² - 2).First, compute 12*(2b² - 2):12*2b² = 24b²,12*(-2) = -24,So, 12*(2b² - 2) = 24b² - 24.Now, subtract that from 16b²:Δ = 16b² - (24b² - 24) = 16b² - 24b² + 24 = -8b² + 24.So, Δ = -8b² + 24.For two distinct real solutions, Δ > 0:-8b² + 24 > 0.Let me solve this inequality:-8b² + 24 > 0,Subtract 24 from both sides:-8b² > -24.Divide both sides by -8, remembering to reverse the inequality sign when dividing by a negative number:b² < 3.So, b² < 3 implies that |b| < sqrt(3), which means b is in the interval (-sqrt(3), sqrt(3)).Therefore, the range of b is (-sqrt(3), sqrt(3)).Alright, that seems straightforward. Now, moving on to part (2): when b = 1, calculate the length of segment AB.So, first, let's substitute b = 1 into the quadratic equation we found earlier:3x² + 4bx + 2b² - 2 = 0.Plugging in b = 1:3x² + 4*1*x + 2*(1)² - 2 = 0,Simplify:3x² + 4x + 2 - 2 = 0,So, 3x² + 4x = 0.Factor out x:x(3x + 4) = 0.Therefore, the solutions are x = 0 and 3x + 4 = 0 => x = -4/3.So, the x-coordinates of points A and B are x1 = 0 and x2 = -4/3.Now, let's find the corresponding y-coordinates using the line equation y = x + b, where b = 1.For x1 = 0:y1 = 0 + 1 = 1.For x2 = -4/3:y2 = (-4/3) + 1 = (-4/3) + (3/3) = (-1/3).So, the coordinates of points A and B are:A: (0, 1),B: (-4/3, -1/3).Now, to find the length of segment AB, I can use the distance formula:|AB| = sqrt[(x2 - x1)² + (y2 - y1)²].Plugging in the values:x1 = 0, y1 = 1,x2 = -4/3, y2 = -1/3.Compute the differences:x2 - x1 = (-4/3) - 0 = -4/3,y2 - y1 = (-1/3) - 1 = (-1/3) - (3/3) = -4/3.So, both differences are -4/3.Now, square these differences:(-4/3)² = 16/9,Similarly, (-4/3)² = 16/9.Add them together:16/9 + 16/9 = 32/9.Take the square root:sqrt(32/9) = (sqrt(32))/3.Simplify sqrt(32):sqrt(32) = sqrt(16*2) = 4*sqrt(2).So, sqrt(32)/3 = (4*sqrt(2))/3.Therefore, the length of segment AB is (4*sqrt(2))/3.Let me just double-check my calculations to make sure I didn't make any mistakes.Starting with the quadratic equation when b=1: 3x² + 4x = 0, which factors to x(3x + 4) = 0, giving x=0 and x=-4/3. That seems correct.Then, plugging into y = x + 1:At x=0, y=1; at x=-4/3, y=-1/3. Correct.Distance formula: sqrt[(-4/3 - 0)^2 + (-1/3 - 1)^2] = sqrt[(16/9) + (16/9)] = sqrt[32/9] = (4*sqrt(2))/3. Yep, that looks right.So, I think I did everything correctly.**Final Answer**(1) The range of ( b ) is (boxed{(-sqrt{3}, sqrt{3})}).(2) The length of segment ( |AB| ) when ( b = 1 ) is (boxed{dfrac{4sqrt{2}}{3}}).

💡Alright, so I've got this geometry problem here about triangle (ABC) with sides (AB = 14), (AC = 15), and (BC = 13). The incircle touches the sides at points (D), (E), and (F), and (I) is the incenter. Then there's this point (X), which is the midpoint of the major arc (widehat{BAC}) of the circumcircle of (ABC). The point (P) is on line (XI) such that (DP) is perpendicular to (EF). I need to find the length of (DP).First off, I should probably draw a diagram to visualize everything. But since I'm just thinking this through, let me note down the key elements:1. Triangle (ABC) with sides 14, 15, 13.2. Incircle with center (I), touching (BC) at (D), (AC) at (E), and (AB) at (F).3. (X) is the midpoint of the major arc (BAC) on the circumcircle.4. (P) is on line (XI) such that (DP perp EF).5. Need to find (DP).Okay, so step one is probably to find some key lengths in the triangle. Maybe the semiperimeter, area, inradius, etc. Let me compute those.The semiperimeter (s) is (frac{14 + 15 + 13}{2} = 21). Then the area (K) can be found using Heron's formula:[K = sqrt{s(s - AB)(s - AC)(s - BC)} = sqrt{21 times 7 times 6 times 8} = sqrt{7056} = 84]So the area is 84. Then the inradius (r) is (K / s = 84 / 21 = 4). Got that, so (r = 4).Now, since (D), (E), and (F) are the points where the incircle touches the sides, I can find the lengths from the vertices to these points. For example, (BD = s - AC = 21 - 15 = 6), (DC = s - AB = 21 - 14 = 7), (AF = AE = s - BC = 21 - 13 = 8), and so on.So, (BD = 6), (DC = 7), (AF = AE = 8), (BF = BD = 6), (CE = CD = 7).Now, (EF) is the segment connecting the points where the incircle touches (AC) and (AB). I need to find the length of (EF). Hmm, how can I find that?I remember that in a triangle, the length of the chord (EF) can be found using the formula:[EF = 2 sqrt{r^2 + left(s - aright)^2}]Wait, no, that doesn't sound quite right. Maybe I should think about coordinates. Let me place the triangle in a coordinate system to make it easier.Let me place point (A) at the origin ((0, 0)), point (B) at ((14, 0)), and point (C) somewhere in the plane. Wait, but with sides 14, 15, 13, that might complicate things. Alternatively, maybe I can use barycentric coordinates or something else.Alternatively, perhaps using trigonometric relationships. Since I know the sides, I can compute the angles of the triangle.Wait, maybe I can compute the coordinates of (E) and (F) and then find the distance between them.Given that (AF = 8) and (AE = 8), so both (E) and (F) are 8 units away from (A) along (AC) and (AB) respectively.But to find the coordinates, I need to know the coordinates of (A), (B), and (C). Maybe I can set (A) at ((0, 0)), (B) at ((c, 0)), and (C) somewhere in the plane. Let me try that.Let me denote:- (A = (0, 0))- (B = (c, 0))- (C = (d, e))But I need to find (c), (d), (e) such that the distances correspond to the given sides.Wait, actually, since (AB = 14), (AC = 15), and (BC = 13), I can set (A) at ((0, 0)), (B) at ((14, 0)), and (C) at ((x, y)). Then, the distance from (A) to (C) is 15, so:[x^2 + y^2 = 15^2 = 225]And the distance from (B) to (C) is 13, so:[(x - 14)^2 + y^2 = 13^2 = 169]Subtracting the first equation from the second:[(x - 14)^2 + y^2 - (x^2 + y^2) = 169 - 225][x^2 - 28x + 196 + y^2 - x^2 - y^2 = -56][-28x + 196 = -56][-28x = -252][x = 9]Then, plugging back into (x^2 + y^2 = 225):[81 + y^2 = 225][y^2 = 144][y = 12 quad (text{since it's above the x-axis})]So, coordinates are:- (A = (0, 0))- (B = (14, 0))- (C = (9, 12))Great, now I can find coordinates for (E) and (F). Since (E) is on (AC) and (AF = 8), but wait, (AF) is on (AB), right? Wait, no, (AF = AE = 8). Wait, no, actually, (AF = s - BC = 8), which is along (AB), and (AE = s - BC = 8) as well, which is along (AC). So, point (F) is 8 units from (A) along (AB), and point (E) is 8 units from (A) along (AC).Since (AB) is from ((0, 0)) to ((14, 0)), moving 8 units along (AB) from (A) is straightforward: (F = (8, 0)).Similarly, (AC) is from ((0, 0)) to ((9, 12)). To find point (E) which is 8 units from (A) along (AC), I can parametrize (AC).The length of (AC) is 15, so the unit vector in the direction of (AC) is ((9/15, 12/15) = (3/5, 4/5)). So, moving 8 units from (A) along (AC) gives:[E = A + 8 times (3/5, 4/5) = (24/5, 32/5) = (4.8, 6.4)]So, (E = (24/5, 32/5)) and (F = (8, 0)).Now, I can find the equation of line (EF). Let's compute the slope first.Slope (m_{EF}):[m_{EF} = frac{0 - 32/5}{8 - 24/5} = frac{-32/5}{(40/5 - 24/5)} = frac{-32/5}{16/5} = -2]So, the slope of (EF) is (-2). Therefore, the equation of line (EF) is:[y - 32/5 = -2(x - 24/5)]Simplifying:[y = -2x + 48/5 + 32/5 = -2x + 80/5 = -2x + 16]So, equation of (EF) is (y = -2x + 16).Now, I need to find point (D), which is the touch point on (BC). Since (BD = 6) and (DC = 7), and (BC) is from ((14, 0)) to ((9, 12)). Let me find the coordinates of (D).First, the length of (BC) is 13, which is consistent with the given sides. So, (D) divides (BC) in the ratio (BD:DC = 6:7).Using the section formula, coordinates of (D):[D = left( frac{6 times 9 + 7 times 14}{6 + 7}, frac{6 times 12 + 7 times 0}{6 + 7} right) = left( frac{54 + 98}{13}, frac{72 + 0}{13} right) = left( frac{152}{13}, frac{72}{13} right)]So, (D = left( frac{152}{13}, frac{72}{13} right)).Now, I need to find point (P) on line (XI) such that (DP perp EF). So, first, I need to find the coordinates of (X), the midpoint of the major arc (BAC) of the circumcircle.Hmm, finding (X) might be a bit tricky. Let me recall that the midpoint of the major arc (BAC) is the ex circumcircle point opposite to (A), or something like that. Alternatively, it's the center of the circle through (B) and (C) that doesn't contain (A). Wait, no, actually, it's the midpoint of the arc (BAC), which is the arc that doesn't contain the opposite side.Alternatively, in triangle (ABC), the midpoint of the major arc (BAC) is the point (X) such that (X) is equidistant from (B) and (C) and lies on the circumcircle.Wait, perhaps I can parametrize the circumcircle and find the midpoint of the major arc.First, let me find the circumradius (R) of triangle (ABC). The formula for circumradius is:[R = frac{abc}{4K} = frac{14 times 15 times 13}{4 times 84}]Calculating numerator: (14 times 15 = 210), (210 times 13 = 2730).Denominator: (4 times 84 = 336).So, (R = 2730 / 336). Simplify:Divide numerator and denominator by 42: (2730 ÷ 42 = 65), (336 ÷ 42 = 8). So, (R = 65/8 = 8.125).So, the circumradius is (65/8).Now, to find the coordinates of (X), the midpoint of the major arc (BAC). Since (X) is the midpoint of the major arc, it's the point such that (AX) is the angle bisector of the external angle at (A), but I'm not sure.Alternatively, in the circumcircle, the midpoint of the major arc (BAC) is the point such that it's the center of the circle passing through (B) and (C) and lies opposite to (A). Wait, perhaps I can find it by reflecting the incenter or something else.Alternatively, maybe it's easier to use complex numbers or parametric equations for the circumcircle.Given that I have coordinates for (A), (B), and (C), I can find the circumcircle equation.The circumcircle passes through (A(0,0)), (B(14,0)), and (C(9,12)). Let me find its equation.The general equation of a circle is (x^2 + y^2 + Dx + Ey + F = 0).Plugging in (A(0,0)):[0 + 0 + 0 + 0 + F = 0 implies F = 0]So, equation becomes (x^2 + y^2 + Dx + Ey = 0).Plugging in (B(14,0)):[14^2 + 0 + D times 14 + E times 0 = 0 implies 196 + 14D = 0 implies 14D = -196 implies D = -14]Plugging in (C(9,12)):[9^2 + 12^2 + D times 9 + E times 12 = 0 implies 81 + 144 - 14 times 9 + 12E = 0][225 - 126 + 12E = 0 implies 99 + 12E = 0 implies 12E = -99 implies E = -99/12 = -33/4]So, the equation of the circumcircle is:[x^2 + y^2 - 14x - frac{33}{4}y = 0]To find the midpoint of the major arc (BAC), which is point (X), I need to find the point on this circle such that it's the midpoint of the arc (BAC). Since (X) is the midpoint of the major arc, it's diametrically opposite to the midpoint of the minor arc (BC).Alternatively, since (X) is the midpoint of the major arc (BAC), it lies on the circumcircle and is the ex circumcircle point opposite to (A). Hmm, perhaps I can find it by rotating point (A) around the circumcircle.Alternatively, since (X) is the midpoint of the major arc (BAC), it should lie such that angles (BXC) is equal to half the measure of the major arc (BAC). Wait, maybe I can parametrize it.Alternatively, perhaps I can find the coordinates of (X) by noting that it's the intersection of the angle bisector of angle (BAC) with the circumcircle, but on the opposite side.Wait, actually, the midpoint of the major arc (BAC) is the point where the internal angle bisector of angle (BAC) meets the circumcircle again. Wait, no, actually, the internal angle bisector meets the circumcircle at the midpoint of the minor arc (BC). The external angle bisector would meet it at the midpoint of the major arc (BC).Wait, perhaps I need to compute the external angle bisector of angle (A) and find its intersection with the circumcircle.Given that, let me compute the external angle bisector of angle (A).First, the internal angle bisector of angle (A) divides the opposite side (BC) in the ratio of the adjacent sides, which is (AB:AC = 14:15). But for the external angle bisector, it divides (BC) externally in the ratio (AB:AC = 14:15).So, the external angle bisector will pass through a point (D') on line (BC) such that (BD'/D'C = AB/AC = 14/15). But since it's external, (D') lies outside segment (BC).Given that (BC) is from ((14, 0)) to ((9, 12)), let me compute the coordinates of (D').Let me denote (BD' = 14k) and (D'C = 15k), with (D') lying beyond (C).So, the ratio (BD'/D'C = 14/15). So, using the section formula for external division:[D' = left( frac{14 times 9 - 15 times 14}{14 - 15}, frac{14 times 12 - 15 times 0}{14 - 15} right) = left( frac{126 - 210}{-1}, frac{168 - 0}{-1} right) = left( frac{-84}{-1}, frac{168}{-1} right) = (84, -168)]Wait, that seems way off. Let me check my calculation.Wait, the external division formula is:[D' = left( frac{m cdot x_2 - n cdot x_1}{m - n}, frac{m cdot y_2 - n cdot y_1}{m - n} right)]Where (m = AB = 14), (n = AC = 15), and points (B(14, 0)) and (C(9, 12)).So,[x = frac{14 times 9 - 15 times 14}{14 - 15} = frac{126 - 210}{-1} = frac{-84}{-1} = 84][y = frac{14 times 12 - 15 times 0}{14 - 15} = frac{168 - 0}{-1} = -168]So, (D' = (84, -168)). That seems correct, but it's quite far from the triangle.Now, the external angle bisector of angle (A) passes through (A(0,0)) and (D'(84, -168)). So, the slope of this bisector is:[m = frac{-168 - 0}{84 - 0} = frac{-168}{84} = -2]So, the equation of the external angle bisector is (y = -2x).Now, to find point (X), which is the intersection of this external angle bisector with the circumcircle, other than (A).So, substituting (y = -2x) into the circumcircle equation:[x^2 + (-2x)^2 - 14x - frac{33}{4}(-2x) = 0][x^2 + 4x^2 - 14x + frac{66}{4}x = 0][5x^2 - 14x + 16.5x = 0][5x^2 + 2.5x = 0][x(5x + 2.5) = 0]So, solutions are (x = 0) (which is point (A)) and (x = -2.5 / 5 = -0.5).So, (x = -0.5), then (y = -2(-0.5) = 1).Therefore, point (X) is at ((-0.5, 1)).Wait, that seems a bit odd, but let me verify.Given that (X) is on the circumcircle and on the external angle bisector, which we found to be (y = -2x). Plugging (x = -0.5), (y = 1) into the circumcircle equation:[(-0.5)^2 + (1)^2 - 14(-0.5) - frac{33}{4}(1) = 0.25 + 1 + 7 - 8.25 = 0.25 + 1 + 7 - 8.25 = 0]Yes, that works. So, (X = (-0.5, 1)).Now, I need to find point (P) on line (XI) such that (DP perp EF).First, let me find the coordinates of (I), the incenter.The incenter coordinates can be found using the formula:[I = left( frac{aA_x + bB_x + cC_x}{a + b + c}, frac{aA_y + bB_y + cC_y}{a + b + c} right)]Wait, no, actually, it's weighted by the lengths of the sides opposite the vertices.Wait, the formula is:[I_x = frac{a cdot A_x + b cdot B_x + c cdot C_x}{a + b + c}][I_y = frac{a cdot A_y + b cdot B_y + c cdot C_y}{a + b + c}]Where (a = BC = 13), (b = AC = 15), (c = AB = 14).So,[I_x = frac{13 cdot 0 + 15 cdot 14 + 14 cdot 9}{13 + 15 + 14} = frac{0 + 210 + 126}{42} = frac{336}{42} = 8][I_y = frac{13 cdot 0 + 15 cdot 0 + 14 cdot 12}{42} = frac{0 + 0 + 168}{42} = 4]So, incenter (I = (8, 4)).Now, line (XI) connects points (X(-0.5, 1)) and (I(8, 4)). Let me find the parametric equation of line (XI).First, compute the direction vector from (X) to (I):[vec{XI} = (8 - (-0.5), 4 - 1) = (8.5, 3)]So, parametric equations:[x = -0.5 + 8.5t][y = 1 + 3t]Where (t) ranges from 0 to 1 to cover the segment (XI), but since (P) is on line (XI), (t) can be any real number.Now, point (P) is somewhere on this line, so its coordinates are ((-0.5 + 8.5t, 1 + 3t)).Now, I need to find (t) such that (DP perp EF). Since (EF) has a slope of (-2), the line perpendicular to (EF) has a slope of (1/2).So, the line (DP) must have a slope of (1/2).Given that (D = left( frac{152}{13}, frac{72}{13} right)) and (P = (-0.5 + 8.5t, 1 + 3t)), the slope of (DP) is:[m_{DP} = frac{(1 + 3t) - frac{72}{13}}{(-0.5 + 8.5t) - frac{152}{13}} = frac{1 + 3t - frac{72}{13}}{-0.5 + 8.5t - frac{152}{13}}]Simplify numerator and denominator:Numerator:[1 - frac{72}{13} + 3t = frac{13}{13} - frac{72}{13} + 3t = frac{-59}{13} + 3t]Denominator:[-0.5 - frac{152}{13} + 8.5t = -frac{1}{2} - frac{152}{13} + frac{17}{2}t]Convert to common denominators:Numerator: (-59/13 + 3t)Denominator:[-frac{13}{26} - frac{304}{26} + frac{221}{26}t = frac{-317}{26} + frac{221}{26}t]So, slope (m_{DP}):[frac{-59/13 + 3t}{-317/26 + 221/26 t} = frac{(-59 + 39t)/13}{(-317 + 221t)/26} = frac{(-59 + 39t) times 26}{13 times (-317 + 221t)} = frac{2(-59 + 39t)}{-317 + 221t}]Set this equal to (1/2):[frac{2(-59 + 39t)}{-317 + 221t} = frac{1}{2}]Cross-multiplying:[4(-59 + 39t) = -317 + 221t][-236 + 156t = -317 + 221t][-236 + 317 = 221t - 156t][81 = 65t][t = 81 / 65 ≈ 1.246]So, (t = 81/65). Now, plug this back into the parametric equations for (P):[x = -0.5 + 8.5 times (81/65) = -0.5 + (17/2) times (81/65) = -0.5 + (1377/130)][Convert -0.5 to -65/130:][x = -65/130 + 1377/130 = (1377 - 65)/130 = 1312/130 = 656/65 ≈ 10.092]Similarly,[y = 1 + 3 times (81/65) = 1 + 243/65 = (65 + 243)/65 = 308/65 ≈ 4.738]So, (P = (656/65, 308/65)).Now, I need to compute the distance (DP). (D = (152/13, 72/13)), (P = (656/65, 308/65)).First, convert (D) to 65 denominator:[D = (152/13, 72/13) = (152 times 5 / 65, 72 times 5 / 65) = (760/65, 360/65)]So, (D = (760/65, 360/65)), (P = (656/65, 308/65)).Compute the differences:[Delta x = 656/65 - 760/65 = (656 - 760)/65 = (-104)/65 = -1.6][Delta y = 308/65 - 360/65 = (308 - 360)/65 = (-52)/65 = -0.8]So, the distance (DP) is:[sqrt{(-1.6)^2 + (-0.8)^2} = sqrt{2.56 + 0.64} = sqrt{3.2} = sqrt{16/5} = 4/sqrt{5} = (4sqrt{5})/5]So, (DP = frac{4sqrt{5}}{5}).**Final Answer**boxed{dfrac{4sqrt{5}}{5}}