Package of problems

## Problem

In this resource, the aim is to understand a fundamental proof of Pythagoras’s Theorem.

### Warm-ups

These first three questions are “warm-ups”, designed to introduce the idea. You are welcome to make use of Pythagoras’s Theorem as you answer them.

In the following questions, the right-angled triangle has the specific side lengths $6$, $8$ and $10$. Would your answers still be correct if we replaced them with other side lengths, say $a$, $b$ and $c$?

1. In the following figure, the red (dark grey if you are using a black-and-white printed version), blue (medium grey) and green (light grey) regions are all semicircles. How are their areas related?

2. In the following figure, the red, blue and green regions are all equilateral triangles. How are their areas related?

3. In the following figure, the red, blue and green regions are all similar triangles. How are their areas related?

### The proof of Pythagoras’s Theorem

In this final question, you are asked to use the ideas you have learnt in the earlier questions to prove Pythagoras’s Theorem. You may not assume Pythagoras’s Theorem to be true when you answer this one!

1. In the left figure below, the blue and green regions have been formed by dropping a perpendicular as shown. How are their areas related to the red area in the right figure?

How does this result prove Pythagoras’s Theorem?

If you have looked at Approach 1 in Proving Pythagoras, this series of problems may have given you an alternative way of thinking about that approach and Pythagoras’s theorem in general.