Question

diagonal \\(\overline{pr}\\) divides rectangle \\(pqrs\\) into two congruent triangles as shown below.

what is the length of \\(\overline{pr}\\) in feet?

\\(\circ\\) \\(4\sqrt{3}\\)
\\(\circ\\) \\(6\sqrt{3}\\)
\\(\circ\\) \\(8\sqrt{3}\\)
\\(\circ\\) \\(12\sqrt{3}\\)

(image description: rectangle \\(pqrs\\) with right angles at \\(s\\) and \\(q\\), diagonal \\(pr\\) creating a \\(30^\circ\\) angle at \\(r\\) with side \\(rq\\) labeled 12 ft.)

Explanation:

Step1: Use cosine function

In right - triangle \(PQR\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), adjacent side \(PQ = 12\) ft, and hypotenuse \(PR\).
So, \(\cos30^{\circ}=\frac{PQ}{PR}\).

Step2: Substitute the value of \(\cos30^{\circ}\)

We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\).
Substituting into the formula \(\frac{\sqrt{3}}{2}=\frac{12}{PR}\).

Step3: Solve for \(PR\)

Cross - multiply: \(PR\times\sqrt{3}=2\times12\).
\(PR=\frac{24}{\sqrt{3}}\).
Rationalize the denominator: \(PR=\frac{24\sqrt{3}}{3}=8\sqrt{3}\).

Answer:

\(8\sqrt{3}\) (the third option)