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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

What a clever little piece of mathematics this is. It is a much neater proof of Pythagoras Theorem than the one I was shown at school. There were also some very well laid out solutions with clear explanations, so well done.

Complete solutions were received from:

Michael - (home educated),
Andrei (School number 205, Bucharest),
Charles (Shrewsbury House School),
Aftab - whose solution is given below.

First Proof

Proof Area of Trapezium derived from square = Area of Trapezium as a sum of areas of three triangles. $${a^2\over{2}}+ ab + {b^2\over{2}} = {b + {c^2\over{2}}}$$ (subtracting ab from both sides) $${{a^2\over{2}}+ ab + {b^2\over{2}}- ab} = {c^2\over{2}}$$ (multiplying 2 both sides) $${a^2 + b^2} = {c^2}$$ (Pythagoras Theorem)

Here is Aftab's solution:

Area of the square = (a+b) 2 (square of sides a+b) $$\mbox{Area of Square} = {{a^2 + 2ab + b^2}}$$

Area of the Trapezium = Area of square divided by 2 (rotational symmetry) $$\mbox{Area of Trapezium} = {{a^2 + 2ab + b^2}\over{2}}$$ $$\mbox{Area of Trapezium} = {{a^2\over{2}}+ ab + {b^2\over{2}}}$$

Area of Trapezium as a sum of areas of triangles $$\mbox{Area of Trapezium} = {{ab\over{2}}+ {ab\over{2}}+ {c^2\over{2}}}$$ $$\mbox{Area of Trapezium} = {{ab + ab + c^2\over{2}}}$$ $$\mbox{Area of Trapezium} = {{2ab + c^2\over{2}} }$$ $$\mbox{Area of Trapezium} = {ab +{ c^2\over{2}} }$$