# Difference between revisions of "2015 AMC 10A Problems/Problem 17"

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We now have the coordinates of two vertices, <math>(1, -\frac{\sqrt{3}}{3})</math> and <math>(1, 1 + \frac{\sqrt{3}}{3})</math>. The length of one side is the distance between the y-coordinates, or <math>1 + \frac{2\sqrt{3}}{3}</math>. | We now have the coordinates of two vertices, <math>(1, -\frac{\sqrt{3}}{3})</math> and <math>(1, 1 + \frac{\sqrt{3}}{3})</math>. The length of one side is the distance between the y-coordinates, or <math>1 + \frac{2\sqrt{3}}{3}</math>. | ||

− | The perimeter of the triangle is thus <math>3 | + | The perimeter of the triangle is thus <math>3(1 + \frac{2\sqrt{3}}{3})</math>, so the answer is <math>\boxed{\textbf{(D) }3 + 2\sqrt{3}}</math> |

## Revision as of 18:32, 4 February 2015

## 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