Question

find the length x.
right triangle with hypotenuse 7.5, vertical leg 5; below the triangle, a segment of length 3 and a right triangle with hypotenuse x, vertical leg 3 (the two triangles are similar)

Explanation:

Step1: Identify similar triangles

The two right triangles are similar by AA similarity (shared angle + right angles).

Step2: Set up proportion

Corresponding sides are proportional: $\frac{7.5}{x} = \frac{5}{3}$

Step3: Solve for x

Cross-multiply and isolate x:
$5x = 7.5 \times 3$
$5x = 22.5$
$x = \frac{22.5}{5}$

Answer:

$4.5$