Library / PMO National Stage Oral Phase /5 of 24

PMO National Stage Oral Phase

Geometry Difficulty 4.4 AIME Prove it Philippines

Problem:

Let $\square ABCD$ be a trapezoid with parallel sides $AB$ and $CD$ of lengths 6 units and 8 units, respectively. Let $E$ be the point of intersection of the extensions of the nonparallel sides of the trapezoid. If the area of $\triangle BEA$ is 60 square units, what is the area of $\triangle BAD$ ?

Solution

Solution:

Note that $\triangle BEA \sim \triangle CED$ and $|EB| = \frac{2 \cdot 60}{6} = 20$ . Thus, $|EC| = \frac{8}{6} \cdot 20 = \frac{80}{3}$ , and hence $|BC| = \frac{80}{3} - 20 = \frac{20}{3}$ . Finally then,
$\text{area of } \triangle BAD = \frac{1}{2} \cdot 6 \cdot \frac{20}{3} = 20$

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.