Question

find the length of side x in simplest radical form with a rational denominator. triangle image: right-angled, two 45° angles, one leg is √3, the other leg is x

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two angles are \(45^{\circ}\), one is \(90^{\circ}\)), so the legs are equal, and the hypotenuse \(h\) and leg \(l\) have the relationship \(h = l\sqrt{2}\), or \(l=\frac{h}{\sqrt{2}}\). Here, the hypotenuse is \(\sqrt{3}\), and \(x\) is a leg.

Step2: Rationalize the Denominator

We know that \(x=\frac{\sqrt{3}}{\sqrt{2}}\). To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{2}\): \(x = \frac{\sqrt{3}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{6}}{2}\).

Answer:

\(\frac{\sqrt{6}}{2}\)