06/17/2020 (2011: National Sprint Round, Problem 30)

Q: In triangle ABC, AB = 12 units and AC = 9 units. Point D is on segment BC so that BD: DC = 2:1. If AD = 6 units, what is the length of segment BC?

A: 3√(33)

First we draw a diagram and label all the lengths that we know and we get:

We use Stewart's Theorem, which basically says...

...and we get: (2x)(3x)(x) + (6)(3x)(6) = (9)(2x)(9) + (12)(x)(12)

That simplifies to:

6x³ + 108x = 162x + 144x -->

6x³ = 198x -->

x³ = 33x -->

x² = 33 -->

x = √(33)

We are trying to find the length of BC, which is x + 2x = 3x. That means it is 3 * 3√(33) = 3√(33).