Question

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

Explanation:

Step1: Analyze the equilateral triangle

In an equilateral triangle, all sides are equal, and the height (the perpendicular from a vertex to the opposite side) splits the triangle into two 30 - 60 - 90 right triangles. The hypotenuse of each right triangle is equal to the side length of the equilateral triangle, which is 10 here. The side opposite the 30° angle is half of the hypotenuse, and the side opposite the 60° angle (which is \(x\) in this case) is \(\frac{\sqrt{3}}{2}\) times the hypotenuse.

Step2: Calculate the length of \(x\)

We know that in a 30 - 60 - 90 triangle, the side opposite 60° (let's call it \(a\)), the hypotenuse (\(c\)) has the relationship \(a=\frac{\sqrt{3}}{2}c\). Here, \(c = 10\), so we substitute \(c\) into the formula:
\(x=\frac{\sqrt{3}}{2}\times10\)
Simplify the expression:
\(x = 5\sqrt{3}\)

Answer:

$5\sqrt{3}$