Question

the triangle below is equilateral. find the length of side x in simplest radical form with a rational denominator.
image of an equilateral triangle with an altitude of length 5, splitting it into two right triangles, and one side labeled x

Explanation:

Step1: Analyze the equilateral triangle

In an equilateral triangle, the altitude (the perpendicular from a vertex to the opposite side) also bisects the side and the angle. So, the right triangle formed (with height 5, hypotenuse \(x\), and base \(\frac{x}{2}\)) is a 30 - 60 - 90 triangle.

Step2: Use trigonometric ratios or Pythagorean theorem

We know that in a 30 - 60 - 90 triangle, \(\sin(60^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\). The angle opposite the height (5) is \(60^{\circ}\), so \(\sin(60^{\circ})=\frac{5}{x}\). Since \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{5}{x}\).

Step3: Solve for \(x\)

Cross - multiply: \(x\times\sqrt{3}=2\times5 = 10\). Then \(x=\frac{10}{\sqrt{3}}\). To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\): \(x=\frac{10\sqrt{3}}{3}\).

Answer:

$\frac{10\sqrt{3}}{3}$