The throne

A rectangle with a horizontal line and a diagonal. The red and golden regions have the same area. What fraction is coloured?

Scroll down for a solution to this problem.


A fraction of 5/9 is coloured.

First, draw the vertical line as shown. Now from equality of red and gold we can fill in the areas A and B. Looking at the B-triangle. Since it has equal area as the rectangle, it must have twice its height. So we know the horizontal line is at 1/3 of the height.

Because of the diagonal, we can deduce that the vertical line is at 1/3 of the rectangle width from the right. This gives A=hw/18. So the little triangle above the diagonal is 1/18 of the rectangle. Added to the half rectangle below the diagonal gives a coloured fraction of 5/9.

Visual solution

Solution without words by David Andriana.


The Iron Throne
In a Game of Throne
With it’s legendary myth
Like the White Walker
Against the Children from the Forest
During the Long Black Night
A fight for the divine Crown
And to keep the Seven Kingdoms
It’s a story for succession of War
For the Iron Throne


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