Question

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

Explanation:

Step1: Use the property of equilateral triangle

In an equilateral triangle, the altitude divides the triangle into two \(30 - 60-90\) right - triangles. The side opposite the \(30^{\circ}\) angle is half of the hypotenuse. Let the side of the equilateral triangle be \(s\). The altitude \(h = 5\). For a \(30 - 60-90\) triangle with hypotenuse \(s\) and shorter leg \(x\) (the side adjacent to the \(60^{\circ}\) angle), the relationship between the sides of a \(30 - 60-90\) triangle is \(h=\frac{\sqrt{3}}{2}s\). But in our case, if we consider the right - triangle formed, and we know that \(\sin60^{\circ}=\frac{5}{x}\).

Step2: Solve for \(x\)

Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), and \(\sin60^{\circ}=\frac{5}{x}\), we have \(\frac{\sqrt{3}}{2}=\frac{5}{x}\). Cross - multiply gives \(x\times\sqrt{3}=10\). Then \(x = \frac{10}{\sqrt{3}}\). Rationalize the denominator: \(x=\frac{10\sqrt{3}}{3}\).

Answer:

\(\frac{10\sqrt{3}}{3}\)