Question

given that △abc is equilateral, and ad bisects ∠a, show that the side lengths of △adb are in a ratio of 1 : √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 = 30°. 3. equilateral triangles are dropdown: right, obtuse, isosceles triangles. image of triangle abc with ad as angle bisector and altitude, right angle at d

Explanation:

An equilateral triangle has all three sides equal and all three angles equal (each \(60^\circ\)). By definition, an isosceles triangle has at least two sides equal (or at least two angles equal). In an equilateral triangle, all three sides are equal, so it satisfies the condition of an isosceles triangle (since having three equal sides means it has at least two equal sides). A right triangle has one \(90^\circ\) angle (and equilateral triangles have \(60^\circ\) angles, so not right), and an obtuse triangle has one angle greater than \(90^\circ\) (equilateral triangles have all angles \(60^\circ\), so not obtuse). So equilateral triangles are isosceles.

Answer:

isosceles