, 2021

Geometry Difficulty 4.5 AIME Prove it United States

Problem:
Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$ . Points $P$ and $Q$ are on segments $EF$ and $CF$ , respectively, such that $A$ , $P$ , and $Q$ are collinear. Given that $EP = 5$ , $PF = 3$ , and $QF = 12$ , find $CQ$ .

Solution

Solution:

Triangles $PFQ$ and $PEA$ are similar, so $AE = FQ \cdot \frac{PE}{PF} = 12 \cdot \frac{5}{3} = 20$ . Now, $CQ = CF - QF = 20 - 12 = 8$ .

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