Question

the triangle below is equilateral. find the length of side ( x ) in simplest radical form with a rational denominator. answer attempt 2 out of 2 ( x = )

Explanation:

Step1: Analyze the properties of an equilateral triangle

An equilateral triangle has all sides equal and each angle is \(60^{\circ}\). When we draw a perpendicular from a vertex to the opposite side (as in the given figure), it bisects the side. So the length of the half - side adjacent to \(x\) is \(\frac{7}{2}\).

Step2: Use trigonometric ratios

We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). In a right - triangle formed (by the perpendicular, half - side and the side \(x\) of the equilateral triangle), \(\theta = 30^{\circ}\) (since the perpendicular bisects the \(60^{\circ}\) angle of the equilateral triangle), the adjacent side to the \(30^{\circ}\) angle is \(\frac{7}{2}\), and the hypotenuse is \(x\).
We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), and \(\cos30^{\circ}=\frac{\frac{7}{2}}{x}\)
So, \(x\times\cos30^{\circ}=\frac{7}{2}\)
Substitute \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\) into the equation: \(x\times\frac{\sqrt{3}}{2}=\frac{7}{2}\)
Multiply both sides by \(\frac{2}{\sqrt{3}}\) to solve for \(x\): \(x = \frac{7}{\sqrt{3}}\)
Rationalize the denominator: \(x=\frac{7\sqrt{3}}{3}\)

Answer:

\(x = \frac{7\sqrt{3}}{3}\)