Question

in the right triangle, there are segments labeled 20, 25, and a side labeled x (with right angles marked). we need to find the value of x.

Explanation:

Step1: Recall the geometric mean theorem (altitude-on-hypotenuse theorem) for right triangles. In a right triangle, the length of a leg is the geometric mean of the length of the hypotenuse segment adjacent to that leg and the length of the entire hypotenuse. The formula is \( x = \sqrt{20\times(20 + 25)} \)? Wait, no, wait. Wait, actually, the two smaller right triangles are similar to the original right triangle and to each other. So, for the leg \( x \), the adjacent segment on the hypotenuse is 20, and the other segment is 25? Wait, no, the hypotenuse is \( 20 + 25 = 45 \)? Wait, no, looking at the diagram, the two segments on the hypotenuse are 20 and 25, and the leg is \( x \). So by the geometric mean theorem, \( x^2 = 20\times(20 + 25) \)? Wait, no, no. Wait, the geometric mean theorem states that in a right triangle, the square of a leg is equal to the product of the hypotenuse and the adjacent segment. Wait, let's correct: If we have a right triangle with hypotenuse \( c \), and an altitude \( h \) to the hypotenuse, dividing the hypotenuse into segments \( a \) and \( b \), then \( a^2 = c\times a \)? No, wait, the correct formula is: for a right triangle, if the altitude to the hypotenuse is drawn, then each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So, if the hypotenuse is split into segments of length \( m \) and \( n \), and the legs are \( p \) and \( q \), then \( p^2 = m\times(m + n) \)? No, wait, no. Let's denote the original right triangle with hypotenuse \( AB = m + n \), altitude \( CD \) to hypotenuse, with \( AD = m \), \( DB = n \), and legs \( AC = p \), \( BC = q \). Then, by similar triangles, \( \triangle ACD \sim \triangle ABC \), so \( \frac{AC}{AB} = \frac{AD}{AC} \), so \( AC^2 = AB\times AD \). So in this problem, the leg \( x \) (let's say \( AC = x \)), \( AD = 20 \), \( AB = 20 + 25 = 45 \)? Wait, no, wait the diagram: the two segments on the hypotenuse are 20 and 25, so the hypotenuse is \( 20 + 25 = 45 \). Then the leg \( x \) is adjacent to the segment 20? Wait, no, maybe I got the segments wrong. Wait, looking at the diagram, the altitude is drawn, creating two segments: 20 and 25. So the hypotenuse is \( 20 + 25 = 45 \). Then the leg \( x \) is such that \( x^2 = 20\times(20 + 25) \)? Wait, no, that can't be. Wait, no, the correct formula is \( x^2 = 20\times(20 + 25) \)? Wait, no, let's check again. Wait, the geometric mean theorem: in a right triangle, the square of a leg is equal to the product of the hypotenuse and the segment of the hypotenuse adjacent to that leg. So if the leg is \( x \), the adjacent segment is 20, and the hypotenuse is \( 20 + 25 = 45 \), then \( x^2 = 20\times 45 \)? Wait, no, that would be \( x^2 = 900 \), so \( x = 30 \). Wait, but let's verify. Alternatively, maybe the two segments are 20 and 25, and the leg is \( x \), so \( x^2 = 20\times(20 + 25) \)? Wait, no, maybe I mixed up the segments. Wait, another way: the two smaller right triangles are similar. So the triangle with leg \( x \) and segment 20 is similar to the triangle with leg \( x \) and hypotenuse \( 20 + 25 \). Wait, no, the original triangle has hypotenuse \( 20 + 25 = 45 \), and one leg is \( x \), the other leg can be found by \( \sqrt{45^2 - x^2} \), but maybe the altitude is \( h \), and \( h^2 = 20\times 25 \), but that's for the altitude. Wait, the problem is to find \( x \), the leg. So by the geometric mean theorem, \( x^2 = 20\times(20 + 25) \)? Wait, no, let's do it step by step.

Wait, the diagram shows a right triangle with an altitude drawn to the hypotenuse, splitting the hypotenuse into segments of 20 and 25. So the hypotenuse length is \( 20 + 25 = 45 \). Let the legs be \( x \) and \( y \), and the altitude be \( h \). Then, by the geometric mean theorem:

- \( x^2 = 20\times 45 \) (because \( x \) is a leg, adjacent to segment 20, and the hypotenuse is 45)
- \( y^2 = 25\times 45 \)
- \( h^2 = 20\times 25 \)

So let's compute \( x^2 = 20\times 45 = 900 \), so \( x = \sqrt{900} = 30 \). Wait, that makes sense.

Step2: Calculate \( x^2 = 20\times(20 + 25) \)

First, compute \( 20 + 25 = 45 \)
Then, \( 20\times 45 = 900 \)
Then, \( x = \sqrt{900} = 30 \)

Answer:

\( \boxed{30} \)