### Why do this problem?

This
problem shows an approach to proving Pythagoras' Theorem by
equating areas calculated in different ways, giving the opportunity
to practise calculating areas. There is historical interest to this
proof as it was discovered by James A Garfield who went on to
become President of the United States.

### Possible approach

Show the image of the trapezium made up of two identical
right-angled triangles joined as shown in the problem. Ask students
to visualise the shape formed by rotating a copy of the trapezium
about the centre point marked. They could sketch the resulting
shape and share their answers in pairs. Then rotate the trapezium
to verify their answers.

Discuss ways of working out the area of the trapezium, both by
calculating areas of squares and right-angled triangles. Set each
pair to work the area out in two different ways, one by finding the
area of the square and halving it, and the other by summing the
areas of the three right-angled triangles.

Each pair can be given the challenge, once they have two
expressions for the same area, of producing a poster or
presentation to explain how this leads to Pythagoras' Theorem. The
lesson could end with each pair presenting their proof to the rest
of the class.

### Key questions

How can we be sure that the green triangle is a right-angled
triangle?

What shape is made by rotating the trapezium around the centre
of the longest side?

How could we work out the area of the trapezium? Is there more
than one way?

### Possible extension

Explore other proofs of Pythagoras in

this
problem.
### Possible support

Tilted
Squares offers another way to think about proving Pythagoras's
Theorem.