Difference between revisions of "2015 AMC 10A Problems/Problem 17"
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− | Since the triangle is equilateral and one of the sides is a vertical line, the other two sides will have opposite slopes. The slope | + | 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 <math>\frac{\sqrt{3}}{3}</math> so the third must be <math>-\frac{\sqrt{3}}{3}</math>. Since this third line passes through the origin, its equation is simply <math>y = -\frac{\sqrt{3}}{3}x</math>. To find two vertices of the triangle, plug in <math>x=1</math> to both the other equations. |
<math>y = -\frac{\sqrt{3}}{3}</math> | <math>y = -\frac{\sqrt{3}}{3}</math> |
Revision as of 18:20, 4 February 2015
Problem
A line that passes through the origin in tersects 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