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Draw a rectangle with sides that are horizontal and vertical, and which is taller than it is wide.

Pick any point on the top side of the rectangle (except either corner).

This point is one of the vertices of a second rectangle. The other vertices of the rectangle all lie on sides of the original rectangle. In fact, there are two rectangles that you can draw.

What is the link between the areas of the three rectangles you have drawn?

There are lots of ways to solve this (including a very nasty, algebraic method). I like this geometric version.

Start by drawing a circle that has its centre at the middle of the rectangle and which passes through the point on the top edge of the rectangle.

This gives a way of constructing the two coloured rectangles. The angles in a rectangle are right angles, which gives us the angle in a semi-circle, so the vertices of the rectangles must be at the intersections of the circle with the original rectangle.

Now focus on the pink quadrilateral. It is made up of half the blue rectangle plus half the green rectangle (with no overlap). This is therefore half the total area of the blue and the green rectangles.

Look at the dotted line. Because it joins two of the points where the circle intersects the original rectangle, we know this line is horizontal. It is now easy to show that the pink quadrilateral is half the area of the original rectangle.

This leads to the result that the areas of the two coloured rectangles sum to the area of the original rectangle.

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