2015 AMC 10A Problems/Problem 17
Contents
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 1
Since the triangle is equilateral and one of the sides is a vertical line, the triangle must have a horizontal line of symmetry, and therefore 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
Solution 2
Draw a line from the y-intercept of the equation perpendicular to the line
. There is a square of side length 1 inscribed in the equilateral triangle. The problems becomes reduced to finding the perimeter of an equilateral triangle with a square of side length 1 inscribed in it. The side length is
. After multiplying the side length by 3 and rationalizing, you get
.
See Also
Video Solution:
https://www.youtube.com/watch?v=2kvSRL8KMac
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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