Solving Triangles by Reflection

ladder against wall

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).

Equilateral Triangle The angles in an
equilateral triangle
are all 60°

 

So the angle between the ladder and the wall is half of 60º

= 30º

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 30º.

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

These examples show that the same triangle can occur in many different situations!