Difference between revisions of "2015 AMC 10A Problems/Problem 17"
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<math>y = 1 + \frac{\sqrt{3}}{3}</math> | <math>y = 1 + \frac{\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>. | + | 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 of the y-coordiantes, or <math>1 + \frac{2\sqrt{3}}{3}</math>. |
− | + | 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> | |
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− | The perimeter of the triangle is <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 19:25, 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 of the y-coordiantes, or
.
The perimeter of the triangle is thus , so the answer is