Question

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

Explanation:

Step1: Identify the triangle type

This is a right - isosceles triangle (45 - 45 - 90 triangle) divided into two congruent right - isosceles triangles. Let the equal sides (excluding the hypotenuse of length 5) be \(a\). Using the Pythagorean theorem \(a^{2}+a^{2}=5^{2}\), \(2a^{2}=25\), \(a^{2}=\frac{25}{2}\), \(a = \frac{5}{\sqrt{2}}\). Rationalizing the denominator, \(a=\frac{5\sqrt{2}}{2}\).

Step2: Use the property of 45 - 45 - 90 triangle

In a 45 - 45 - 90 triangle, if the length of the leg is \(l\), the length of the hypotenuse \(x\) (for the smaller right - isosceles triangle) is related by \(x=\sqrt{2}l\). Here \(l = \frac{5\sqrt{2}}{2}\). Then \(x=\sqrt{2}\times\frac{5\sqrt{2}}{2}\).

$$x=\frac{2\times5}{2}= \frac{5\sqrt{2}}{2}$$

Answer:

\(\frac{5\sqrt{2}}{2}\)