Question

the triangle below is equilateral. find the length of side $x$ in simplest radical form with a rational denominator.
answer attempt 1 out of 2
$x = 11$

Explanation:

Step1: Use properties of equilateral triangle

In an equilateral triangle, the altitude bisects the base. Let the side of the equilateral triangle be \(a = 11\). After drawing the altitude, we get a right - triangle with hypotenuse \(a = 11\) and one of the acute angles \(\theta=60^{\circ}\) (original angle of equilateral triangle) and the side adjacent to the \(30^{\circ}\) (since \(90 - 60=30\)) angle is \(x\).

Step2: Apply cosine function in right - triangle

We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\theta = 30^{\circ}\) and hypotenuse \(c = 11\), \(\cos30^{\circ}=\frac{x}{11}\). Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have the equation \(\frac{\sqrt{3}}{2}=\frac{x}{11}\).

Step3: Solve for \(x\)

Cross - multiply the equation \(\frac{\sqrt{3}}{2}=\frac{x}{11}\) to get \(x=\frac{11\sqrt{3}}{2}\).

Answer:

\(x = \frac{11\sqrt{3}}{2}\)