Question

use special right triangles to find the missing side length
find the length of side x in simplest radical form with a rational denominator.
(image of a right triangle with a 45° angle, a right angle, the leg opposite the 45° angle is √8, and the hypotenuse is x)

Explanation:

Step1: Identify Triangle Type

This is a 45-45-90 triangle (two 45° angles, one right angle). In such triangles, legs are equal, hypotenuse \( = \text{leg} \times \sqrt{2} \). Here, leg length is \( \sqrt{8} \), hypotenuse is \( x \).

Step2: Apply 45-45-90 Formula

For a 45-45-90 triangle, hypotenuse \( x = \text{leg} \times \sqrt{2} \). Substitute leg \( = \sqrt{8} \):
\( x = \sqrt{8} \times \sqrt{2} \)
Simplify \( \sqrt{8} \times \sqrt{2} = \sqrt{16} = 4 \) (since \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \), and \( \sqrt{16} = 4 \)).

Answer:

\( 4 \)