Competition problem — Geometry — Maths Olympiad Prep

Library / /40 of 60

Geometry Difficulty 4.1 AIME Find the answer South Africa

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 $E$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

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 $d^2 = 2 \times 20 = 40$ , so $e^2 = 40 - 16 = 24$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.