Question

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Explanation:

Step1: Identify the geometric theorem

This is a right triangle with an altitude to the hypotenuse, so we can use the geometric mean theorem (or altitude-on-hypotenuse theorem), which states that in a right triangle, the length of a leg is the geometric mean of the length of the hypotenuse and the length of the adjacent segment of the hypotenuse. So, if we let the hypotenuse be \( c = 25 + 20=45 \)? Wait, no, wait. Wait, actually, looking at the diagram, the large triangle is a right triangle, and there's a perpendicular from the right angle to the hypotenuse? Wait, no, the diagram shows a right triangle with one leg divided into 20 and (let's say) the other part, and the hypotenuse segment is 25? Wait, no, maybe I misread. Wait, the triangle has a right angle at the bottom, and there's a perpendicular from the right angle to the hypotenuse? Wait, no, the labels: the side with 20 and 25? Wait, no, the diagram: the hypotenuse of the small right triangle is 25? Wait, no, let's re-express. Let's denote the large right triangle as \( \triangle ABC \) with right angle at \( C \), and a perpendicular \( CD \) to \( AB \), where \( AD = 20 \), \( DB = 25 \)? No, wait, the length labeled 20 and 25: maybe the segment of the hypotenuse is 20, and the other segment is... Wait, no, the problem: the side labeled 20 and 25, and the leg is \( x \). Wait, actually, the correct theorem is that in a right triangle, the square of a leg is equal to the product of the hypotenuse and the adjacent segment. Wait, no, the geometric mean theorem: in a right triangle, if an altitude is drawn to the hypotenuse, then each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Wait, but in this diagram, maybe the large triangle is a right triangle, and the segment of the hypotenuse is 20, and the other part is... Wait, no, the length from the vertex to the foot of the perpendicular is 20, and the hypotenuse segment is 25? Wait, no, maybe the leg is \( x \), and the hypotenuse of the large triangle is \( 20 + 25 = 45 \)? No, that can't be. Wait, maybe I made a mistake. Wait, the correct approach: let's denote the large right triangle with legs \( x \) and \( y \), hypotenuse \( z \), and the altitude to the hypotenuse divides the hypotenuse into segments \( a = 20 \) and \( b = 25 \). Then, by the geometric mean theorem, \( x^2 = a \times (a + b) \)? No, wait, no: the geometric mean theorem states that \( x^2 = a \times z \), where \( z \) is the hypotenuse, and \( a \) is the segment adjacent to \( x \). Wait, no, the formula is: if in right triangle \( \triangle ABC \), right-angled at \( C \), and \( CD \perp AB \), then \( AC^2 = AD \times AB \), \( BC^2 = BD \times AB \), and \( CD^2 = AD \times BD \). So in this case, if \( AD = 20 \), \( AB = 20 + 25 = 45 \)? No, that doesn't make sense. Wait, maybe the length labeled 25 is the hypotenuse of the small triangle? Wait, no, the diagram: the side with 20 and 25, and the leg is \( x \). Wait, maybe the correct equation is \( x^2 = 20 \times (20 + 25) \)? No, that would be \( x^2 = 20 \times 45 = 900 \), so \( x = 30 \). Wait, but let's check. Wait, if the segment of the hypotenuse is 20, and the other segment is 25, then the hypotenuse is \( 20 + 25 = 45 \)? No, that can't be, because 20 and 25 are parts of the leg? Wait, no, I think I misread the diagram. Wait, the correct diagram: the right triangle has a leg divided into 20 and (let's say) the other part, and the hypotenuse segment is 25? No, maybe the length from the vertex to the foot of the perpendicular is 20, and the length of the hypotenuse segment is 25, so the leg \( x \) satisfies \( x^2 = 20 \times (20 + 25) \)? Wait, no, the geometric mean theorem: \( x^2 = AD \times AB \), where \( AD \) is the segment adjacent to \( x \), and \( AB \) is the hypotenuse. Wait, if \( AD = 20 \), and \( AB = 20 + 25 = 45 \), then \( x^2 = 20 \times 45 = 900 \), so \( x = 30 \). Alternatively, maybe the hypotenuse of the large triangle is 25, and the segment is 20? No, that would be \( x^2 = 20 \times 25 = 500 \), which is not a perfect square. Wait, no, the correct diagram: the large right triangle, with a perpendicular from the right angle to the hypotenuse, creating two smaller similar triangles. The segment of the hypotenuse adjacent to the leg \( x \) is 25, and the other segment is 20? Wait, no, the length labeled 20 and 25: maybe the leg of the small triangle is 20, and the hypotenuse is 25? No, that would be a right triangle with legs 20 and \( h \), hypotenuse 25, so \( h = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15 \), but that's not \( x \). Wait, I think I messed up the diagram. Wait, the correct approach: the triangle is a right triangle, and the segment of the hypotenuse is 20, and the other part is 25? No, the problem is to find \( x \), and the sides are 20 and 25. Wait, maybe the formula is \( x^2 = 20 \times (20 + 25) \)? No, that's not right. Wait, no, the geometric mean theorem: in a right triangle, the square of a leg is equal to the product of the hypotenuse and the adjacent segment. Wait, let's denote the hypotenuse as \( c \), and the segment adjacent to leg \( x \) as \( a \), then \( x^2 = a \times c \). If \( a = 20 \), and \( c = 20 + 25 = 45 \), then \( x^2 = 20 \times 45 = 900 \), so \( x = 30 \). Alternatively, if the hypotenuse is 25, and the segment is 20, then \( x^2 = 20 \times 25 = 500 \), which is not a perfect square. So I think the correct diagram is that the hypotenuse of the large triangle is \( 20 + 25 = 45 \), and the segment adjacent to \( x \) is 25? Wait, no, that would be \( x^2 = 25 \times 45 = 1125 \), which is not a perfect square. Wait, I'm confused. Wait, maybe the diagram is a right triangle with one leg divided into 20 and (let's say) the other part, and the hypotenuse is 25? No, that doesn't make sense. Wait, let's look at the numbers: 20, 25, and x. 20-25-30? Wait, 20, 25, and 30: no, 20^2 + 15^2 = 25^2 (since 20-15-25 is a Pythagorean triple). Wait, maybe the triangle is a right triangle with legs 15 and 20, hypotenuse 25, but that's the 3-4-5 triangle scaled by 5. But in the diagram, there's a perpendicular from the right angle to the hypotenuse, creating two smaller triangles. Then, the length of the leg x would be the hypotenuse of the larger small triangle. Wait, the altitude to the hypotenuse in a 20-15-25 triangle: the altitude \( h \) is \( \frac{20 \times 15}{25} = 12 \). Then, the segments of the hypotenuse are \( \frac{20^2}{25} = 16 \) and \( \frac{15^2}{25} = 9 \). But that doesn't match 20 and 25. Wait, maybe the diagram is labeled differently. Wait, the problem says "Find the value of x", and the diagram has a right triangle with a perpendicular from the right angle to the hypotenuse, dividing the hypotenuse into 20 and 25? No, that can't be, because 20 + 25 = 45, and the leg would be \( \sqrt{20 \times 45} = \sqrt{900} = 30 \). Ah, that makes sense. So the hypotenuse is 45, divided into 20 and 25, and the leg x is \( \sqrt{20 \times 45} = 30 \). So that's the solution.

Step2: Apply the geometric mean theorem

Using the geometric mean theorem (leg - segment - hypotenuse relationship): for a right triangle, if a leg is adjacent to a segment of the hypotenuse of length \( a \) and the hypotenuse has length \( c = a + b \), then the leg length \( x \) satisfies \( x^2 = a \times c \). Here, \( a = 20 \), \( c = 20 + 25 = 45 \). So:

\[ x^2 = 20 \times 45 \]
\[ x^2 = 900 \]
\[ x = \sqrt{900} = 30 \]

Answer:

30