Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.9 AIME Prove it United States

Problem:
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$ , $AB = 1$ , and $AC = 2$ . Let $\ell$ be a line through $A$ perpendicular to $BC$ , and let the perpendicular bisectors of $AB$ and $AC$ meet $\ell$ at $E$ and $F$ , respectively. Find the length of segment $EF$ .

Solution

Solution:
Answer: $\frac{3 \sqrt{5}}{4}$
Let $M, N$ be the midpoints of $AB$ and $AC$ , respectively. Then we have $\angle EAB = \angle ACB$ and $\angle EAC = \angle ABC$ , so $AEM \sim CBA \Rightarrow AE = \frac{\sqrt{5}}{4}$ and $FAN \sim CBA \Rightarrow AF = \sqrt{5}$ . Consequently, $EF = AF - AE = \frac{3 \sqrt{5}}{4}$ .

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