Question

find the length of \\( \overline { a b } \\)

Explanation:

Step1: Recall the tangent function

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 30^{\circ}\), the opposite side to \(\theta\) is \(BC = 11\), and the adjacent side is \(AB\). So, \(\tan30^{\circ}=\frac{BC}{AB}\).

Step2: Substitute the value of \(\tan30^{\circ}\)

Since \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\), we have \(\frac{1}{\sqrt{3}}=\frac{11}{AB}\).

Step3: Solve for \(AB\)

Cross - multiply: \(AB = 11\sqrt{3}\).

Answer:

\(11\sqrt{3}\) (the first option)