Question

1. 11√3 30° a b (right triangle diagram)

Explanation:

Step1: Identify triangle type

This is a 30-60-90 right triangle. In such a triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite \(30^\circ\) is the shortest (let's call it \(b\)), the side opposite \(60^\circ\) is \(b\sqrt{3}\) (here, \(11\sqrt{3}\) is opposite \(60^\circ\)), and the hypotenuse \(a\) is \(2b\).

Step2: Find \(b\)

The side \(11\sqrt{3}\) is opposite \(60^\circ\), so \(b\sqrt{3}=11\sqrt{3}\). Divide both sides by \(\sqrt{3}\): \(b = 11\).

Step3: Find \(a\) (hypotenuse)

In a 30-60-90 triangle, hypotenuse \(a = 2b\). Substitute \(b = 11\): \(a = 2\times11 = 22\).

Answer:

\(b = 11\), \(a = 22\)