Question

find the length x. 7.5 x 5 4.5

Explanation:

Step1: Identify Similar Triangles

The two right triangles are similar (by AA similarity, as both have a right angle and share the vertical angle). So, their corresponding sides are proportional.

Step2: Set Up Proportion

Let the sides be proportional: $\frac{x}{4.5} = \frac{5}{7.5}$ (wait, correction: actually, the correct proportion is $\frac{x}{5} = \frac{4.5}{7.5}$? Wait, no. Let's label the triangles. Let the smaller triangle have legs 4.5 and x, hypotenuse? Wait, no, the larger triangle has legs 5 and 7.5? Wait, no, the larger triangle: one leg is 5, the hypotenuse segment? Wait, no, the two triangles: the smaller right triangle with legs 4.5 and x, and the larger right triangle with legs 5 and (let's see, the other leg is 7.5? Wait, no, the hypotenuse of the smaller triangle is part of the hypotenuse of the larger? Wait, no, the two triangles are similar, so corresponding sides: smaller triangle: leg 4.5, leg x; larger triangle: leg 5, leg (let's see, the other leg is 7.5? Wait, no, the larger triangle's leg is 7.5? Wait, maybe the proportion is $\frac{x}{5} = \frac{4.5}{7.5}$? Wait, no, let's do it properly. Let the smaller triangle have legs 4.5 (vertical) and x (horizontal), and the larger triangle have legs 7.5 (vertical) and 5 (horizontal). Wait, no, the right angles: both triangles are right-angled, and they share the angle at the intersection, so they are similar. So corresponding sides: smaller triangle's vertical leg (4.5) to larger triangle's vertical leg (7.5) is equal to smaller triangle's horizontal leg (x) to larger triangle's horizontal leg (5). So proportion: $\frac{4.5}{7.5} = \frac{x}{5}$

Step3: Solve for x

Cross-multiplying: $7.5x = 4.5 \times 5$
$7.5x = 22.5$
Divide both sides by 7.5: $x = \frac{22.5}{7.5} = 3$

Answer:

3