💡Okay, so I have this geometry problem here, and I need to figure out how to prove that lines PQ and AB are parallel in this specific triangle setup. Let me try to break it down step by step.First, the problem says that triangle ABC is such that AB equals BC, and angle ABC is 90 degrees. So, triangle ABC is an isosceles right triangle with AB = BC and the right angle at B. That means sides AB and BC are equal, and the hypotenuse AC must be longer. Since it's a right triangle, I can use the Pythagorean theorem if needed.Next, a height BH is drawn in the triangle. In a right triangle, the altitude from the right angle to the hypotenuse has some special properties. I remember that in a right triangle, the altitude to the hypotenuse creates two smaller triangles that are similar to each other and to the original triangle. So, triangle ABH is similar to triangle CBH, and both are similar to triangle ABC.Now, point P is chosen on side CA such that AP equals AB. Since AB is one of the legs of the triangle, AP is equal in length to AB. That means point P is somewhere on CA such that the segment from A to P is equal to AB. I should probably sketch this to visualize where P is located.Similarly, point Q is chosen on side CB such that BQ equals BH. BH is the altitude from B to AC, so BQ is equal to that length. Since BH is an altitude, it's shorter than BC, so Q is somewhere between B and C on side CB.The goal is to prove that lines PQ and AB are parallel. To show that two lines are parallel, I can use various methods like showing corresponding angles are equal, or using the converse of the basic proportionality theorem (Thales' theorem), or showing that the slopes are equal if I use coordinate geometry.Maybe using coordinate geometry would be a good approach here because it allows me to assign coordinates to each point and calculate the necessary slopes or vectors to show parallelism.Let's assign coordinates to the triangle. Let me place point B at the origin (0,0) since it's the right angle. Since AB = BC, let's assume AB = BC = 1 for simplicity. Then, point A can be at (0,1) and point C can be at (1,0). That way, AB is along the y-axis from (0,0) to (0,1), BC is along the x-axis from (0,0) to (1,0), and AC is the hypotenuse from (0,1) to (1,0).Now, let's find the coordinates of point H, the foot of the altitude from B to AC. The equation of AC can be found since it goes from (0,1) to (1,0). The slope of AC is (0-1)/(1-0) = -1, so the equation is y = -x + 1. The altitude from B to AC is perpendicular to AC, so its slope is the negative reciprocal of -1, which is 1. Since it passes through B(0,0), the equation is y = x.To find point H, we solve the system of equations y = -x + 1 and y = x. Setting them equal: x = -x + 1 => 2x = 1 => x = 1/2. Then y = 1/2. So, H is at (1/2, 1/2).Next, point P is on CA such that AP = AB. AB is 1 unit, so AP should also be 1 unit. Let's find the coordinates of P. Since A is at (0,1) and C is at (1,0), the length of AC is sqrt((1-0)^2 + (0-1)^2) = sqrt(2). So, AP is 1, which is less than sqrt(2), so P is somewhere between A and C.To find P, we can parametrize AC. Let me use a parameter t such that when t=0, we are at A(0,1), and t=1, we are at C(1,0). So, the parametric equations are x = t, y = 1 - t. The distance from A to a general point (t, 1 - t) is sqrt((t - 0)^2 + ((1 - t) - 1)^2) = sqrt(t^2 + t^2) = sqrt(2t^2) = t*sqrt(2). We want this distance to be 1, so t*sqrt(2) = 1 => t = 1/sqrt(2). Therefore, the coordinates of P are (1/sqrt(2), 1 - 1/sqrt(2)).Similarly, point Q is on CB such that BQ = BH. BH is the length from B(0,0) to H(1/2,1/2), which is sqrt((1/2)^2 + (1/2)^2) = sqrt(1/4 + 1/4) = sqrt(1/2) = 1/sqrt(2). So, BQ should be 1/sqrt(2). Since CB is from (0,0) to (1,0), it's along the x-axis. So, moving from B(0,0) towards C(1,0), a distance of 1/sqrt(2) along the x-axis would place Q at (1/sqrt(2), 0).Now, we have coordinates for P and Q. Let me write them down:- P: (1/sqrt(2), 1 - 1/sqrt(2))- Q: (1/sqrt(2), 0)Now, let's find the slope of PQ. The slope is (y2 - y1)/(x2 - x1). Plugging in the coordinates:Slope of PQ = (0 - (1 - 1/sqrt(2))) / (1/sqrt(2) - 1/sqrt(2)) = (-1 + 1/sqrt(2)) / 0.Wait, that's undefined. That would mean PQ is a vertical line. But AB is from (0,0) to (0,1), which is also a vertical line. So, if both PQ and AB are vertical, they are parallel.But wait, let me double-check my calculations because dividing by zero suggests that the x-coordinates are the same, which they are for both P and Q. So, PQ is indeed a vertical line, same as AB, hence they are parallel.Alternatively, if I consider vectors, the vector PQ is (0, - (1 - 1/sqrt(2))), and the vector AB is (0,1). Both vectors are vertical, so they are scalar multiples of each other, confirming that PQ is parallel to AB.Hmm, that seems straightforward. But let me think if there's another way to approach this without coordinate geometry, maybe using similar triangles or properties of the isosceles right triangle.Since ABC is an isosceles right triangle, and BH is the altitude, which we know splits the triangle into two smaller isosceles right triangles, ABH and CBH. So, both ABH and CBH are similar to ABC.Point P is on AC such that AP = AB. Since AB = 1, AP = 1, and AC = sqrt(2), so P divides AC in the ratio AP:PC = 1 : (sqrt(2) - 1). Similarly, Q is on CB such that BQ = BH = 1/sqrt(2). Since CB = 1, BQ = 1/sqrt(2), so QC = 1 - 1/sqrt(2).If I can show that the triangles APB and QPB are similar or something like that, but I'm not sure. Maybe using the basic proportionality theorem, which states that if a line divides two sides of a triangle proportionally, then it is parallel to the third side.Wait, in triangle ABC, if I can show that PQ divides sides AB and BC proportionally, then PQ would be parallel to AC. But in this case, we need to show PQ is parallel to AB, not AC. Maybe I need to consider another triangle.Alternatively, since both PQ and AB are vertical lines in my coordinate system, they are parallel by definition. But maybe in a different coordinate system, this wouldn't hold, but since the triangle is fixed, the conclusion should be general.I think the coordinate approach was the most straightforward here, showing that both PQ and AB are vertical lines, hence parallel. Alternatively, using vectors or slopes also confirms this.Another way is to consider transformations. Since AP = AB and BQ = BH, and given the symmetry of the isosceles right triangle, the points P and Q are constructed in such a way that PQ ends up being a translated version of AB, hence parallel.But perhaps the simplest explanation is the coordinate method, which clearly shows both lines are vertical, thus parallel.So, after going through this, I feel confident that PQ is indeed parallel to AB.

💡First, I'll define the number of turtles Kristen has as ( K ).Kris has one fourth of Kristen's turtles, so Kris has ( frac{K}{4} ) turtles.Trey has five times as many turtles as Kris, which means Trey has ( 5 times frac{K}{4} = frac{5K}{4} ) turtles.The total number of turtles is the sum of Kristen's, Kris's, and Trey's turtles:[K + frac{K}{4} + frac{5K}{4} = 30]Combining the terms:[K + frac{K}{4} + frac{5K}{4} = K + frac{6K}{4} = K + frac{3K}{2} = frac{5K}{2}]Setting this equal to 30:[frac{5K}{2} = 30]Solving for ( K ):[5K = 60 K = 12]So, Kristen has 12 turtles. Kris has ( frac{12}{4} = 3 ) turtles, and Trey has ( frac{5 times 12}{4} = 15 ) turtles.Adding them up:[12 + 3 + 15 = 30]Thus, the total number of turtles is 30.

💡First, I need to determine how far Hannah walked. Let's denote the distance Hannah walked as ( x ) kilometers.According to the problem, Nadia walked twice as far as Hannah, so Nadia walked ( 2x ) kilometers.The total distance walked by both girls is 27 kilometers. Therefore, I can set up the equation:[ x + 2x = 27 ]Combining like terms, the equation becomes:[ 3x = 27 ]To find the value of ( x ), I divide both sides of the equation by 3:[ x = frac{27}{3} ][ x = 9 ]So, Hannah walked 9 kilometers. Since Nadia walked twice as far as Hannah, I multiply Hannah's distance by 2 to find Nadia's distance:[ 2 times 9 = 18 ]Therefore, Nadia walked 18 kilometers.

💡Okay, so I have this problem about a right-angled triangle ABC. The perimeter is 14, and the area is 7. I need to find the lengths of its three sides. Hmm, right-angled triangle, so I know one of the angles is 90 degrees, and the sides satisfy the Pythagorean theorem. Let me think about how to approach this.First, let me denote the sides. Let's say the two legs are 'a' and 'b', and the hypotenuse is 'c'. So, according to the Pythagorean theorem, I have:a² + b² = c² ...(1)That's one equation. Now, the perimeter is 14, so the sum of all sides is 14:a + b + c = 14 ...(2)And the area is 7. Since it's a right-angled triangle, the area is (1/2)*base*height, which in this case would be (1/2)*a*b:(1/2)*a*b = 7 ...(3)So, I have three equations here. I need to solve for a, b, and c. Let me see how I can manipulate these equations to find the values.Starting with equation (3), I can multiply both sides by 2 to get rid of the fraction:a*b = 14 ...(3a)So, the product of a and b is 14. That's useful.Now, looking at equation (2), a + b + c = 14. Maybe I can express c in terms of a and b. Let's rearrange equation (2):c = 14 - a - b ...(2a)So, c is 14 minus the sum of a and b. That makes sense because the perimeter is fixed.Now, let's plug this expression for c into equation (1). So, equation (1) becomes:a² + b² = (14 - a - b)²Let me expand the right-hand side:(14 - a - b)² = 14² - 2*14*(a + b) + (a + b)²= 196 - 28(a + b) + a² + 2ab + b²So, substituting back into equation (1):a² + b² = 196 - 28(a + b) + a² + 2ab + b²Hmm, let's subtract a² + b² from both sides to simplify:0 = 196 - 28(a + b) + 2abSo, 2ab - 28(a + b) + 196 = 0I can divide the entire equation by 2 to make it simpler:ab - 14(a + b) + 98 = 0 ...(4)Now, from equation (3a), I know that ab = 14. Let me substitute that into equation (4):14 - 14(a + b) + 98 = 0Simplify:14 + 98 - 14(a + b) = 0112 - 14(a + b) = 0So, 14(a + b) = 112Divide both sides by 14:a + b = 8 ...(5)Alright, so now I know that a + b = 8 and ab = 14. This looks like a system of equations that I can solve using quadratic equations.Let me consider a and b as the roots of a quadratic equation. If I let x be a variable, then the quadratic equation would be:x² - (a + b)x + ab = 0Substituting the known values:x² - 8x + 14 = 0Now, I can solve this quadratic equation using the quadratic formula:x = [8 ± √(64 - 56)] / 2= [8 ± √8] / 2= [8 ± 2√2] / 2= 4 ± √2So, the two roots are 4 + √2 and 4 - √2. Therefore, the lengths of the two legs are 4 + √2 and 4 - √2.Now, let's find the hypotenuse c. From equation (2a):c = 14 - a - b= 14 - (a + b)= 14 - 8= 6So, the hypotenuse is 6.Let me just verify if these values satisfy all the original equations.First, check the Pythagorean theorem:a² + b² = (4 + √2)² + (4 - √2)²= (16 + 8√2 + 2) + (16 - 8√2 + 2)= 18 + 18= 36And c² = 6² = 36. So, that checks out.Next, check the perimeter:a + b + c = (4 + √2) + (4 - √2) + 6= 4 + √2 + 4 - √2 + 6= 4 + 4 + 6 + √2 - √2= 14That's correct.Finally, check the area:(1/2)*a*b = (1/2)*(4 + √2)*(4 - √2)= (1/2)*(16 - (√2)²)= (1/2)*(16 - 2)= (1/2)*14= 7Perfect, that matches the given area.So, all the conditions are satisfied with the sides being 4 - √2, 4 + √2, and 6.

💡Okay, so I have this problem here: If two different square roots of a positive number are (2m - 6) and (m + 3), then (m) is ____.Alright, let me try to figure this out step by step. First, I know that every positive number has two square roots: one positive and one negative. For example, the square roots of 9 are 3 and -3. So, in this problem, (2m - 6) and (m + 3) must be these two square roots of the same positive number.Since they are square roots of the same number, they must satisfy the property that one is the negative of the other. That is, if one square root is (a), the other should be (-a). So, I can set up an equation where one expression is equal to the negative of the other.Let me write that down:[2m - 6 = -(m + 3)]Hmm, okay. So, I have (2m - 6) equals the negative of (m + 3). Let me solve this equation for (m).First, I'll distribute the negative sign on the right side:[2m - 6 = -m - 3]Now, I want to get all the (m) terms on one side and the constants on the other. Let me add (m) to both sides to get rid of the (-m) on the right:[2m + m - 6 = -3]Simplifying the left side:[3m - 6 = -3]Next, I'll add 6 to both sides to isolate the term with (m):[3m = 3]Now, I'll divide both sides by 3 to solve for (m):[m = 1]Wait, so (m) is 1? Let me check if this makes sense. If (m = 1), then let's compute both square roots:First square root: (2m - 6 = 2(1) - 6 = 2 - 6 = -4)Second square root: (m + 3 = 1 + 3 = 4)Okay, so the two square roots are -4 and 4, which are indeed negatives of each other. That means they are the two square roots of the same positive number, which is 16 because ((-4)^2 = 16) and (4^2 = 16).Let me just make sure I didn't make any mistakes in my calculations. Starting from the equation:[2m - 6 = -(m + 3)]Expanding the right side:[2m - 6 = -m - 3]Adding (m) to both sides:[3m - 6 = -3]Adding 6 to both sides:[3m = 3]Dividing by 3:[m = 1]Yep, that all checks out. So, (m = 1) is the correct answer. I think I've got it!

💡Okay, so I have this problem where I need to use the bisection method to find the real root of the equation (x^3 - 2x - 5 = 0) in the interval ([2, 3]). They've given me that the midpoint of the interval is (x_0 = 2.5), and I need to figure out which of the next intervals contains the root. The options are A: ([2, 2.5]), B: ([2.5, 3]), C: (left[ frac{5}{2}, frac{11}{4}right]), and D: None of the above.Alright, let me start by recalling what the bisection method is. From what I remember, the bisection method is a numerical method used to find roots of a continuous function. It works by repeatedly narrowing down an interval where the function changes sign, meaning it goes from positive to negative or vice versa. The idea is that if a function changes sign over an interval, then by the Intermediate Value Theorem, there must be at least one root in that interval.So, the steps are roughly: 1) Check the function values at the endpoints of the interval. 2) If the function changes sign, then the root is in that interval. 3) Find the midpoint of the interval and evaluate the function there. 4) Determine which half of the interval the root lies in by checking the sign change. 5) Repeat the process on the new interval.Alright, let's apply this to the given equation (f(x) = x^3 - 2x - 5). The initial interval is ([2, 3]). I need to compute (f(2)) and (f(3)) to see if the function changes sign in this interval.Calculating (f(2)):(f(2) = 2^3 - 2*2 - 5 = 8 - 4 - 5 = -1). So, (f(2) = -1), which is negative.Calculating (f(3)):(f(3) = 3^3 - 2*3 - 5 = 27 - 6 - 5 = 16). So, (f(3) = 16), which is positive.Since (f(2)) is negative and (f(3)) is positive, the function changes sign between 2 and 3, so there must be a root in this interval. Good, that's consistent with the problem statement.Now, they've given the midpoint (x_0 = 2.5). Let me compute (f(2.5)) to see where the function is at this midpoint.Calculating (f(2.5)):First, (2.5^3) is (2.5 * 2.5 * 2.5). Let me compute that step by step:- (2.5 * 2.5 = 6.25)- (6.25 * 2.5 = 15.625)So, (2.5^3 = 15.625)Then, (2x) when (x = 2.5) is (2 * 2.5 = 5)So, (f(2.5) = 15.625 - 5 - 5 = 15.625 - 10 = 5.625). So, (f(2.5) = 5.625), which is positive.Wait, so (f(2) = -1) is negative, (f(2.5) = 5.625) is positive. That means the function changed sign between 2 and 2.5, right? So, the root must lie in the interval ([2, 2.5]).But hold on, let me make sure I'm not making a mistake here. The bisection method says that if (f(a)) and (f(c)) have opposite signs, then the root is in ([a, c]), otherwise, it's in ([c, b]). So, in this case, (a = 2), (b = 3), and (c = 2.5).Since (f(a) = -1) is negative and (f(c) = 5.625) is positive, they have opposite signs. Therefore, the root lies in ([2, 2.5]). So, the next interval should be ([2, 2.5]), which is option A.But wait, let me check the options again. Option C is (left[ frac{5}{2}, frac{11}{4}right]). Let me compute what that is in decimal to see if it's the same as ([2, 2.5]).(frac{5}{2} = 2.5) and (frac{11}{4} = 2.75). So, option C is ([2.5, 2.75]). That's different from option A, which is ([2, 2.5]). So, why is option C there?Wait, maybe I made a mistake in interpreting the midpoint. Let me double-check the midpoint calculation. The midpoint of ([2, 3]) is indeed (2.5), so that's correct. Then, evaluating (f(2.5)) gave me a positive value, so the root is between 2 and 2.5, which is option A.But let me make sure that I didn't miscalculate (f(2.5)). Maybe I did that wrong.Calculating (f(2.5)) again:(2.5^3 = 15.625)(2x = 5)So, (f(2.5) = 15.625 - 5 - 5 = 5.625). Yes, that's correct.So, (f(2.5)) is positive, so the root is in ([2, 2.5]). So, the next interval is ([2, 2.5]), which is option A.But wait, why is option C there? Let me think. Maybe the question is asking for the next interval after taking the midpoint, but perhaps they are considering further subdivisions?Wait, no. The bisection method is about taking the midpoint and then deciding which half to keep. So, in the first step, we have ([2, 3]), midpoint at 2.5, evaluate (f(2.5)), which is positive, so since (f(2)) is negative, the root is in ([2, 2.5]). So, the next interval is ([2, 2.5]), which is option A.But let me check the options again. Option C is (left[ frac{5}{2}, frac{11}{4}right]), which is ([2.5, 2.75]). That seems like the next midpoint if we were to continue bisecting, but in this case, we're only asked for the next interval after the first midpoint.Wait, maybe I'm overcomplicating. The question says: "Taking the midpoint of the interval (x_0 = 2.5), the next interval containing the root is (　　)." So, after taking the midpoint, we determine which half to keep. Since (f(2.5)) is positive and (f(2)) is negative, the root is in ([2, 2.5]), so the next interval is ([2, 2.5]), which is option A.But let me make sure I'm not missing something. Maybe the question is considering that the midpoint is 2.5, and the next interval is from 2.5 to the next midpoint? Wait, no, that doesn't make sense. The bisection method is about halving the interval each time, so after the first midpoint, we keep the half where the function changes sign.So, in this case, since (f(2)) is negative and (f(2.5)) is positive, the function changes sign between 2 and 2.5, so the root is in ([2, 2.5]). Therefore, the next interval is ([2, 2.5]), which is option A.But wait, let me check if the function actually has a root in ([2, 2.5]). Maybe I should graph it or compute more points to be sure. Let me compute (f(2.25)) to see.Calculating (f(2.25)):(2.25^3 = 2.25 * 2.25 * 2.25)First, (2.25 * 2.25 = 5.0625)Then, (5.0625 * 2.25 = 11.390625)So, (2.25^3 = 11.390625)Then, (2x = 4.5)So, (f(2.25) = 11.390625 - 4.5 - 5 = 11.390625 - 9.5 = 1.890625), which is positive.So, (f(2.25)) is positive. Since (f(2) = -1) is negative and (f(2.25)) is positive, the root is between 2 and 2.25. So, that's within ([2, 2.5]), so yes, the interval ([2, 2.5]) is correct.Wait, but if I compute (f(2.125)), let's see:(2.125^3 = 2.125 * 2.125 * 2.125)First, (2.125 * 2.125 = 4.515625)Then, (4.515625 * 2.125 ≈ 9.61328125)So, (f(2.125) = 9.61328125 - 4.25 - 5 = 9.61328125 - 9.25 ≈ 0.36328125), which is still positive.So, (f(2.125)) is positive. So, the root is between 2 and 2.125. So, that's still within ([2, 2.5]).Wait, but if I compute (f(2.0625)):(2.0625^3 ≈ 2.0625 * 2.0625 * 2.0625)First, (2.0625 * 2.0625 ≈ 4.251953125)Then, (4.251953125 * 2.0625 ≈ 8.7890625)So, (f(2.0625) ≈ 8.7890625 - 4.125 - 5 ≈ 8.7890625 - 9.125 ≈ -0.3359375), which is negative.Ah, so (f(2.0625)) is negative. So, the function changes sign between 2.0625 and 2.125. So, the root is in ([2.0625, 2.125]). But that's getting into more detailed subdivisions. However, in the context of the bisection method, after the first midpoint, we only need to determine the next interval, which is ([2, 2.5]), as we found earlier.So, going back, the question is asking for the next interval after taking the midpoint (x_0 = 2.5). Since (f(2.5)) is positive and (f(2)) is negative, the root is in ([2, 2.5]), so the next interval is ([2, 2.5]), which is option A.But wait, let me check the options again. Option C is (left[ frac{5}{2}, frac{11}{4}right]), which is ([2.5, 2.75]). That seems like the next interval if we were to take the midpoint of ([2.5, 3]), but in this case, since the root is in ([2, 2.5]), we don't need to consider ([2.5, 3]) anymore.Wait, but maybe I'm misunderstanding the question. It says, "Taking the midpoint of the interval (x_0 = 2.5), the next interval containing the root is (　　)." So, after taking the midpoint, we have two intervals: ([2, 2.5]) and ([2.5, 3]). We evaluate (f(2.5)) and find it's positive, so since (f(2)) is negative, the root is in ([2, 2.5]). Therefore, the next interval is ([2, 2.5]), which is option A.But let me make sure that I'm not confusing the next interval with the next midpoint. The next interval is simply the half where the function changes sign, which is ([2, 2.5]). So, the answer should be A.Wait, but let me think again. Maybe the question is considering that after taking the midpoint, the next interval is the one that includes the midpoint and the next point? No, that doesn't make sense. The bisection method is about halving the interval each time, so after the first midpoint, we keep the half where the function changes sign.So, in this case, since (f(2)) is negative and (f(2.5)) is positive, the root is in ([2, 2.5]), so the next interval is ([2, 2.5]), which is option A.But wait, let me check if the function actually has a root in ([2, 2.5]). I've already computed (f(2.5) = 5.625) and (f(2) = -1), so yes, the function changes sign there. Therefore, the root is indeed in ([2, 2.5]).So, the correct answer should be A.Wait, but let me check the options again. Option C is (left[ frac{5}{2}, frac{11}{4}right]), which is ([2.5, 2.75]). That's the interval from 2.5 to 2.75, which is the right half of the original interval. But since the function at 2.5 is positive and at 3 is positive, the function doesn't change sign in ([2.5, 3]), so the root isn't there. Therefore, option C is incorrect.So, the correct answer is A: ([2, 2.5]).Wait, but let me think about the initial function. (f(x) = x^3 - 2x - 5). Let me see if I can find an approximate root to confirm.Using the Newton-Raphson method for a quick approximation, starting with (x_0 = 2):(f(2) = -1)(f'(x) = 3x^2 - 2)(f'(2) = 12 - 2 = 10)Next approximation: (x_1 = 2 - (-1)/10 = 2 + 0.1 = 2.1)Compute (f(2.1)):(2.1^3 = 9.261)(2*2.1 = 4.2)(f(2.1) = 9.261 - 4.2 - 5 = 0.061), which is very close to zero.So, the root is approximately 2.1, which is within ([2, 2.5]). Therefore, the interval containing the root is indeed ([2, 2.5]), so option A is correct.But wait, let me check if the function actually crosses zero between 2 and 2.5. Since (f(2) = -1) and (f(2.5) = 5.625), and the function is continuous, by the Intermediate Value Theorem, there must be a root in ([2, 2.5]). So, that confirms it.Therefore, the correct answer is A: ([2, 2.5]).