Competition problem — Geometry — Maths Olympiad Prep

Library / /17 of 20

Geometry Difficulty 4.9 AIME Prove it South Africa

In the figure, $ABCD$ is a rectangle.
The area of triangle $DEC$ is $9\ \text{cm}^2$ and $BE = \frac{2}{5} BC$ .
What is the area of $ABCD$ in $\text{cm}^2$ ?

Solution

Let $BE = 2x$ ; then with $BE = \frac{2}{5} BC$ we must have $EC = 3x$ .
Now for $\triangle DEC$ , $\frac{1}{2}(3x)(DC) = 9$ , giving $DC = \frac{6}{x}$ , and since $BC = 5x$ ,
the area of rectangle $ABCD$ is $5x \cdot \frac{6}{x} = 30$ .

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.