Question

find the length x.

Explanation:

Step1: Use the Pythagorean theorem for the first triangle.

Let the first right - triangle have sides \(a = 5\), \(b\) (unknown) and hypotenuse \(c = 7.5\). By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), so \(b=\sqrt{c^{2}-a^{2}}=\sqrt{7.5^{2}-5^{2}}=\sqrt{(7.5 + 5)(7.5 - 5)}=\sqrt{12.5\times2.5}=\sqrt{\frac{25}{2}\times\frac{5}{2}}=\sqrt{\frac{125}{4}}=\frac{5\sqrt{5}}{2}\)

Step2: Since the two triangles are similar.

The ratios of corresponding sides are equal. The ratio of the vertical to the horizontal side in the first triangle is the same as in the second triangle. The two right - triangles are similar, so \(\frac{5}{b}=\frac{2}{x}\). Substitute \(b = \frac{5\sqrt{5}}{2}\) into \(\frac{5}{b}=\frac{2}{x}\), we get \(5x = 2b\). Then \(x=\frac{2b}{5}\). Substitute \(b=\frac{5\sqrt{5}}{2}\) into it, \(x=\sqrt{5}\)

Answer:

\(\sqrt{5}\)