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Dividing the Field

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two trapeziums each of equal area. How could he do this?

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Same Height

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?

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Major Trapezium

A small circle in a square in a big circle in a trapezium. Using the measurements and clue given, find the area of the trapezium.

Garfield's Proof

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.