Question

find the length of side ( x ) in simplest radical form with a rational denominator.

Explanation:

Step1: Identify the triangle type

The triangle has two \(45^\circ\) angles and a right angle, so it's a 45 - 45 - 90 triangle. In a 45 - 45 - 90 triangle, the legs are equal, and the hypotenuse \(c\) is related to the leg \(a\) by \(c=a\sqrt{2}\). Here, the leg length is 4, and \(x\) is the hypotenuse.

Step2: Apply the 45 - 45 - 90 triangle formula

For a 45 - 45 - 90 triangle, hypotenuse \(=\) leg \(\times\sqrt{2}\). Given leg \( = 4\), so \(x = 4\sqrt{2}\).

Answer:

\(4\sqrt{2}\)