Difference between revisions of "2015 AMC 10A Problems/Problem 17"
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==Problem== | ==Problem== | ||
− | A line that passes through the origin | + | A line that passes through the origin intersects both the line <math> x = 1</math> and the line <math>y=1+ \frac{\sqrt{3}}{3} x</math>. The three lines create an equilateral triangle. What is the perimeter of the triangle? |
<math> \textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)} \ 2+2\sqrt{3} \qquad\textbf{(C)} \ 6 \qquad\textbf{(D)} \ 3 + 2\sqrt{3} \qquad\textbf{(E)} \ 6 + \frac{\sqrt{3}}{3} </math> | <math> \textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)} \ 2+2\sqrt{3} \qquad\textbf{(C)} \ 6 \qquad\textbf{(D)} \ 3 + 2\sqrt{3} \qquad\textbf{(E)} \ 6 + \frac{\sqrt{3}}{3} </math> | ||
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==Solution== | ==Solution== |
Revision as of 18:20, 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 . Apply the distance formula, .
The length of one side is
The perimeter of the triangle is , so the answer is