# Two equilateral triangles

Prove that these two lengths are equal.

## Problem

The two black triangles are both equilateral, and their bases form a straight line segment.

Prove that the two red lines are of equal length.

Image

## Student Solutions

**Using rotation**

Consider the coloured triangles in the diagram below. They both have one side equal in length to the larger equilateral triangle, and these sides are at an angle of 60$^\text{o}$ to each other. They also both have one side equal in length to the smaller equilateral triangle, and these sides are also at an angle of 60$^\text{o}$ to each other.

Image

So a 60$^\text{o}$ rotation will map the blue triangle onto the red triangle. So the two triangles must be congruent. So the third sides, which are the red lengths, must be equal.

**Using 'side angle side'**

Let the sides of the larger equilateral triangle have length $a$ and the sides of the smaller equilateral triangle have length $b$, as labeled in the diagram. Notice that both the red triangle and the blue triangle contain an angle of 120$^\text{o}$ between sides of length $a$ and $b$.

Image

The lengths of the two sides and the size of the angle in between them uniquely define a triangle (imagine putting the ends of two sticks together at a fixed angle, joining the other ends of the sticks would complete the triangle). So the two triangles must be congruent, which means the third lengths must also be the same.