South African Mathematics Olympiad

Question

Five squares (labelled A, B, C, D and E) are drawn along a straight line, each touching the next square at a vertex, as shown in the diagram. Square B has an area of 20. Square C has a side length of 4. Square D has double the area of square B. What is the area of square E? FIGURE_0

Accepted Answer

Denote the side lengths of squares A, B, C, D, E by a, b, c, d, e respectively. The two right-angled triangles enclosed by squares C, D, E and the line are congruent to each other and have sides c, d, e, so by Pythagoras' theorem d^2 = c^2 + e^2. We are given that c^2 = 4^2 = 16 and d^2 = 2 20 = 40, so e^2 = 40 - 16 = 24.