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 (can be find out easily by using 60 degree angle of the triangle) 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)
Let the intersection point between the line and the line that crosses the origin be .
We drop an altitude from onto the line . Since the overall triangle is an equilateral triangle, we are splitting the base (on ) in half. As the y-axis is parallel to the line , the altitude from P will also split the y-axis from to in half. From this, we can get that the y-value of P is .
Plugging this into the equation , we get that , and thus our height for the equilateral triangle is . Using that, we can calculate the perimeter to be .
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