Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.1 AIME Find the answer United States

Problem:

Triangle $A B C$ is given in the plane. Let $A D$ be the angle bisector of $\angle B A C$ ; let $B E$ be the altitude from $B$ to $A D$ , and let $F$ be the midpoint of $A B$ . Given that $A B=28$ , $B C=33$ , $C A=37$ , what is the length of $E F$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

$14$

$\triangle A B E$ is a right triangle, and $F$ is the midpoint of the hypotenuse (and therefore the circumcenter), so $E F = B F = A F = 14$ .

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