Question

find the value of x .

(image of a large right triangle and a smaller similar right triangle inside it. the large triangle has a vertical leg of 5 m, horizontal leg from t to v of 9 m, and the smaller triangle has a vertical leg of x m, horizontal leg from v to w of 6 m.)

x = \square

Explanation:

Step1: Use similarity of triangles

Since the two right - angled triangles are similar (by AA similarity, as they have a common angle at \(W\) and both have a right angle), the ratios of their corresponding sides are equal.
The ratio of the bases is \(\frac{9}{9 + 6}=\frac{9}{15}=\frac{3}{5}\), and the ratio of the heights should be the same.

Step2: Set up the proportion

Let the height of the smaller triangle be \(x\) and the height of the larger triangle be \(5\). Then \(\frac{x}{5}=\frac{9}{9 + 6}\).
Substitute the values: \(\frac{x}{5}=\frac{9}{15}\).
Cross - multiply: \(15x=5\times9\).
Simplify: \(15x = 45\).
Solve for \(x\): \(x=\frac{45}{15}\).

Answer:

\(3\)