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?
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?
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.
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.