Question

the triangle below is equilateral. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Analyze the angles

Since the triangle is equilateral, each angle is \(60^{\circ}\). The smaller right - triangle has an angle of \(30^{\circ}\) (because the angle of the equilateral triangle is bisected). In a \(30 - 60-90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). Let the side opposite \(30^{\circ}\) be \(a\), the side opposite \(60^{\circ}\) be \(a\sqrt{3}\), and the hypotenuse be \(2a\). Here, the side opposite \(30^{\circ}\) is \(x\), and the side opposite \(60^{\circ}\) is \(5\).

Step2: Use the ratio of \(30 - 60-90\) triangle

We know that for a \(30 - 60-90\) triangle, if the side opposite \(60^{\circ}\) is \(y=a\sqrt{3}\) and the side opposite \(30^{\circ}\) is \(x = a\). Given \(y = 5=a\sqrt{3}\), then \(a=\frac{5}{\sqrt{3}}\).

Step3: Rationalize the denominator

Rationalize \(\frac{5}{\sqrt{3}}\) by multiplying numerator and denominator by \(\sqrt{3}\). So \(x=\frac{5\sqrt{3}}{3}\).

Answer:

\(\frac{5\sqrt{3}}{3}\)