Question

given that △abc is equilateral, and overline{ad} bisects ∠a, show that the side lengths of △adb are in a ratio of 1 : sqrt{3} : 2.

1. equilateral triangles are also equiangular, so m∠a = m∠b = m∠c = 60°.
2. by definition of an angle bisector, m∠bad = m∠cad = dropdown with 30°, 45°, 60°

image of triangle abc with ad as an angle bisector and altitude, d on bc with right angle at d

Explanation:

Step1: Recall angle in equilateral triangle

In an equilateral triangle, all interior angles are equal. Since the sum of interior angles of a triangle is \(180^{\circ}\), each angle \(m\angle A = m\angle B = m\angle C=\frac{180^{\circ}}{3} = 60^{\circ}\).

Step2: Determine angle after bisecting

\(\overline{AD}\) bisects \(\angle A\), so it divides \(\angle A\) (which is \(60^{\circ}\)) into two equal angles. So \(m\angle BAD=m\angle CAD = \frac{60^{\circ}}{2}=30^{\circ}\).

Answer:

\(30^{\circ}\)