2015 AMC 10A Problems/Problem 17
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?
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
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 problem 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 .
Solution 3 (Intuitive)
With the first condition, we have that
Then, we have
Dividing both sides by on the second and putting over a common denominator gets us The only answer in the answer choices that satisfies this is (D)
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