Question

find the missing length indicated.

Explanation:

Step1: Recall the geometric mean theorem (altitude-on-hypotenuse theorem) for right triangles. In a right triangle, the length of a leg (x) is the geometric mean of the length of the hypotenuse segment adjacent to that leg and the length of the entire hypotenuse? Wait, no, actually, when an altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Wait, let's correct: The geometric mean theorem (or altitude-on-hypotenuse theorem) states that in a right triangle, the altitude drawn to the hypotenuse is the geometric mean between the segments into which it divides the hypotenuse. Also, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Wait, looking at the diagram: The triangle is an isoceles? No, wait, the big triangle is a right triangle? Wait, the diagram shows a triangle with a right angle at the top, and an altitude drawn to the base (hypotenuse) of length 100, dividing it into a segment of 36 and the other segment (let's call it y, so 36 + y = 100, so y = 64). Wait, no, the leg is x, the altitude is h, and the hypotenuse is 100, with one segment being 36. Wait, according to the geometric mean theorem, for a right triangle, if we have a leg x, and the hypotenuse is 100, and the adjacent segment to x is 36? Wait, no, the formula is: If a right triangle has hypotenuse c, and is divided into two segments a and b (a + b = c) by the altitude to the hypotenuse, then each leg (let's say leg l) satisfies \( l^2 = a \times c \)? Wait, no, let's rederive.

Let the right triangle be \( \triangle ABC \) with right angle at \( C \), and altitude \( CD \) to hypotenuse \( AB \), where \( D \) is on \( AB \). Then \( AD = 36 \), \( AB = 100 \), so \( DB = 100 - 36 = 64 \). Then, by the geometric mean theorem, \( AC^2 = AD \times AB \)? Wait, no, \( AC^2 = AD \times AB \)? Wait, no, actually \( AC^2 = AD \times AB \) is incorrect. Wait, the correct formula is \( AC^2 = AD \times AB \)? Wait, no, let's use similar triangles. \( \triangle ACD \sim \triangle ABC \) (by AA similarity, since both are right triangles and share angle \( A \)). Therefore, the ratios of corresponding sides are equal: \( \frac{AC}{AB} = \frac{AD}{AC} \), so \( AC^2 = AD \times AB \). Wait, but in the diagram, the leg is x (which is \( AC \)), \( AD = 36 \), \( AB = 100 \). Wait, but that would give \( x^2 = 36 \times 100 \), but that would be \( x = \sqrt{3600} = 60 \). Wait, but let's check the other segment. Wait, if \( AD = 36 \), \( AB = 100 \), then \( DB = 64 \), and \( BC^2 = DB \times AB = 64 \times 100 = 6400 \), so \( BC = 80 \). But the options include 60 and 80. Wait, the problem is to find the missing length indicated, which is x. Wait, the diagram shows x as the leg, with the adjacent segment 36. Wait, maybe I mixed up. Wait, maybe the hypotenuse is 100, and the segment adjacent to x is 36, so \( x^2 = 36 \times 100 \)? No, that would be 60. Wait, but let's check the options: 60 is an option (green button). Wait, let's confirm.

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 hypotenuse is 100, and the adjacent segment is 36, then \( x^2 = 36 \times 100 \)? Wait, no, that would be if the segment is adjacent to the leg, but actually, when you draw the altitude to the hypotenuse, the two smaller triangles are similar to the original triangle and to each other. So \( \triangle ACD \sim \triangle ABC \), so \( \frac{AC}{AB} = \frac{AD}{AC} \), so \( AC^2 = AD \times AB \). So \( AC = \sqrt{AD \times AB} = \sqrt{36 \times 100} = \sqrt{3600} = 60 \). So x = 60.

Step2: Calculate \( x \) using the geometric mean theorem.

Given \( AD = 36 \), \( AB = 100 \), and \( AC = x \). By the geometric mean theorem (similar triangles), \( x^2 = AD \times AB \).
Substitute the values: \( x^2 = 36 \times 100 = 3600 \).
Take the square root of both sides: \( x = \sqrt{3600} = 60 \).

Answer:

60 (corresponding to the green button with 60)