Solving Triangles by Reflection

A 5ft ladder leans against a wall as shown. What is the angle between the ladder and the wall?

This is surprisingly easy to solve by using Reflection:

Here is the triangle with its reflection Together they make an equilateral triangle (all sides equal).

The angles in an equilateral triangle are all 60°

So the angle between the ladder and the wall is half of 60° = 30°

Finding Length

We can use the same idea to find an unknown length.

Alex has a laser that measures distance.

By standing some distance from the tree Alex measures 42m to the top of the tree at an angle of 30°.

What is the height of the tree?

Here is the triangle and its reflection:

Once again the triangle and its reflection make an equilateral triangle.

So, we know the height of the tree must be half of 42m

= 21m

In fact reflection can help in many cases!