Question

-1/1 points given that the pair of triangles is similar, find the length y. (let a = 36 and b = 16.)

Explanation:

Step1: Identify geometric mean theorem

For similar triangles, the geometric mean theorem (altitude-on-hypotenuse theorem) applies: \( y^2 = a \times b \) (wait, no—wait, the segments: actually, when a triangle is divided by an angle bisector or altitude, but here, since the triangles are similar, and the sides are divided proportionally. Wait, actually, the correct relation for the geometric mean (altitude to hypotenuse in a right triangle, but here, the triangles are similar, so the ratio of corresponding sides. Wait, let's re-express: if the two triangles are similar, and the segments on the side are such that \( y \) is the geometric mean of \( a \) and \( b \)? Wait, no, the problem: given \( a = 36 \), \( b = 16 \), and the triangles are similar, so by the geometric mean theorem (also called the leg rule or altitude-on-hypotenuse), in a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. Wait, but the diagram: the triangles are similar, so the ratio of sides. Wait, actually, the formula is \( y = \sqrt{a \times b} \)? Wait, no, wait: if the two triangles are similar, and the sides are \( a \), \( b \), and \( y \), then by the geometric mean (altitude theorem), \( y^2 = a \times b \)? Wait, no, let's check: if we have a right triangle, and an altitude to the hypotenuse, then the altitude is the geometric mean of the two segments. Wait, but here, the given \( a = 36 \), \( b = 16 \), so \( y = \sqrt{36 \times 16} \)? Wait, no, wait: maybe the triangles are similar, so the ratio of corresponding sides. Wait, the problem says "the pair of triangles is similar", so the sides are proportional. Let's assume that \( y \) is the geometric mean, so \( y^2 = a \times b \). Wait, \( a = 36 \), \( b = 16 \), so \( y = \sqrt{36 \times 16} \).

Step2: Calculate \( y \)

First, compute \( 36 \times 16 \): \( 36 \times 16 = 576 \). Then, take the square root of 576: \( \sqrt{576} = 24 \).

Answer:

\( y = 24 \)