HMMT November, 2018

Geometry Difficulty 4.8 AIME Prove it United States

Problem:

$H O W$ , $B O W$ , and $D A H$ are equilateral triangles in a plane such that $W O = 7$ and $A H = 2$ . Given that $D$ , $A$ , $B$ are collinear in that order, find the length of $B A$ .

Solution

Solution:

Note that $H \neq B$ since otherwise $D A B$ is an equilateral triangle. Let $M$ be the midpoint of $D A$ , so $H B = 7 \sqrt{3}$ and $H M = \sqrt{3}$ , and $\angle H M B = 90^{\circ}$ . By the Pythagorean theorem,
$B M = \sqrt{(7 \sqrt{3})^2 - (\sqrt{3})^2} = 12$
Then $B A = B M - A M = 11$ .

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