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?

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?

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:
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Rotate a copy of the trapezium about the centre of the longest side of the blue triangle to make a square. What is the area of the square? From this formula for the area of this square derive a formula for the area of the trapezium.

Now write down the area of the trapezium as the sum of the areas of the three right angled triangles.

Use these results to give a proof of Pythagoras Theorem explaining each step.

This proof is credited to James A. Garfield (1876) the 20 th President of the United States.