The Deathly Hollow

Geometry Level pending

Triangle ABC is an equilateral triangle circumscribing circle O with point O in the center. An altitude from point B to line AC intersects circle O at point E and line AC at point F. If line BE is 5 3 5{\sqrt3} cm, what is the perimeter of triangle ABC (in cm)?


The answer is 90.

Alex Christopher Estrella by Alex Christopher Estrella , 23

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1 solution

Marta Reece
Jun 15, 2017

A C = a AC=a , B F = 3 2 a BF=\dfrac{\sqrt3}{2}a

O F = 1 3 × 3 2 a = a 2 3 OF=\dfrac13\times\dfrac{\sqrt3}{2}a=\dfrac{a}{2\sqrt3}

B E = B F − 2 × O F = 3 2 a − 2 × a 2 3 = 5 3 BE=BF-2\times OF=\dfrac{\sqrt3}{2}a-2\times\dfrac{a}{2\sqrt3}=5\sqrt3

a = 5 3 3 2 − 1 3 = 30 a=\dfrac{5\sqrt3}{\dfrac{\sqrt3}{2}-\dfrac{1}{\sqrt3}}=30 , 3 a = 90 3a=\boxed{90}

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