# 2015 AMC 10A Problems/Problem 17

## Problem

A line that passes through the origin intersects both the line and the line . The three lines create an equilateral triangle. What is the perimeter of the triangle?

## Solution

Since the triangle is equilateral and one of the sides is a vertical line, the other two sides will have opposite slopes. The slope of the other given line is so the third must be . Since this third line passes through the origin, its equation is simply . To find two vertices of the triangle, plug in to both the other equations.

We now have the coordinates of two vertices, and . The length of one side is the distance between the y-coordinates, or .

The perimeter of the triangle is thus , so the answer is