### Warm-up

### The problem

A square of side $(a+b)$ has a smaller square of side $c$ inscribed.

Find the area of the big square, the small square and the four triangles in terms of $a$, $b$, and $c$. Use these areas to proof pythagorus' theorem.

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### Solution:

The area of one of the triangles is

$${\mathrm{Area}}_{\text{Triangle}}=\mathrm{\xbd}ab$$The area of the large square is

$${\mathrm{Area}}_{\text{Large}}={(a+b)}^{2}$$and for the small

$${\mathrm{Area}}_{\text{Small}}={c}^{2}$$Using the diagram to aid combining these areas

$${\mathrm{Area}}_{\text{Large}}={\mathrm{Area}}_{\text{Small}}+4\times {\mathrm{Area}}_{\text{Triangle}}$$ $${(a+b)}^{2}={c}^{2}+4\times \mathrm{\xbd}ab$$ $${a}^{2}+2ab+{b}^{2}={c}^{2}+2ab$$ $${a}^{2}+{b}^{2}={c}^{2}$$As required

### Further Reading:

Pythagorus' Theorem

(Care of thirdspacelearning.com)

GCSE Maths Proof - Revision and Worksheets

(care of mmerevise.co.uk)