Question

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

Explanation:

Step1: Identify the triangle type

This is a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle. In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the ratio of the sides is \(1:1:\sqrt{2}\). Let the legs be \(a\) and \(a\), and the hypotenuse be \(c = a\sqrt{2}\). Here, \(c = 2\), and we need to find \(a=x\).

Step2: Use the ratio formula

From \(c=a\sqrt{2}\), we can solve for \(a\). Substitute \(c = 2\) into the formula: \(2=a\sqrt{2}\). Then \(a=\frac{2}{\sqrt{2}}\).

Step3: Rationalize the denominator

Multiply numerator and denominator by \(\sqrt{2}\): \(\frac{2\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{2\sqrt{2}}{2}=\sqrt{2}\).

Answer:

\(\sqrt{2}\)