# Mock AIME 4 2006-2007 Problems/Problem 11

## Problem

Let be an equilateral triangle. Two points and are chosen on and , respectively, such that . Let be the intersection of and . The area of is 13 and the area of is 3. If , where , , and are relatively prime positive integers, compute .

## Solution

Let , and . Note that we want to compute the ratio .

Assign a mass of to point . This gives point a mass of and point a mass of . Thus, .

Since the ratio of areas of triangles that share an altitude is simply the ratio of their bases, we have that:

.

By the quadratic formula, we find that , so .

Thus, our final answer is .