# Matter of Scale

## Problem

Take any right-angled triangle with side lengths $a, b$ and $c$. For convenience, label the two acute angles $x^{\circ}$ and $y^{\circ}$.

Make two enlargements of the triangle, one by scale factor $a$ and and one by scale factor $b$:

**What are the lengths of these new triangles?**

Draw out a copy of them and indicate what the lengths and the angles are in each.

Draw out a copy of them and indicate what the lengths and the angles are in each.

Find the lengths and angles in this last triangle.**Can you show that this triangle is similar to the original triangle?**

What is the scale factor of enlargement between the first and last triangles?

**Can you use your results to prove Pythagoras' Theorem?**

*You might like to explore some more proofs of Pythagoras' Theorem, and a proof of The Converse of Pythagoras' Theorem.*

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*

## Getting Started

The red triangle is an enlargement of the original by a scale factor of $a$. The lengths of its sides are marked on.

Can you mark on the lengths of the blue triangle?

Which sides match together to make a larger triangle?

What can you say about the angles of the resulting triangle when the red and blue triangle are put together?

What is the scale factor of enlargement from the original triangle to the last triangle?

## Student Solutions

Well done to Nayanika from The Tiffin Girls' School, Yihuan form Pate's Grammar School, John from Calthorpe Park School in the UK and Andrew from Island School, who all sent in correct proofs.

This is John's work:

The four triangles are similar by AA similarity *(they all share two angles - and therefore all 3 angles)*. As triangle 4 is similar to triangle 1 its corresponding parts are in equal ratios.

$\therefore a:ac$ is the same ratio as $c:a^2+b^2$

$\therefore 1:c$ $=$ $c:a^2+b^2$

$\therefore 1:c$ $=$ $1:\dfrac{a^2+b^2}{c}$

$\therefore c = \dfrac{a^2+b^2}{c}$

Which is rearranged to $a^2+b^2=c^2$

Alternatively we can think of triangle 4 as triangle 1 enlarged by a scale factor of $c$.

Therefore $a^2+b^2 = c\times c$

So $a^2+b^2=c^2$

## Teachers' Resources

This problem uses enlargements, scale factors and similar triangles to create a proof of Pythagoras' theorem.

This problem can be used alongside other proofs of Pythagoras' theorem, and students can consider which ones they think are most convincing, and which are easiest to understand/explain.

Students can also investigate a proof of the Converse of Pythagoras.