2005 AMC 12B Problems/Problem 24
Problem
All three vertices of an equilateral triangle are on the parabola , and one of its sides has a slope of . The -coordinates of the three vertices have a sum of , where and are relatively prime positive integers. What is the value of ?
Solution
Let the points be , and . Using elementary calculus concepts and the fact that they lie on ,
= the slope of ,
= the slope of ,
= the slope of .
So the value that we need to find is simply the sum of the slopes of the three sides of the triangle divided by . WLOG, let be the side that has the smallest angle with the positive -axis. Let be an arbitrary point with the coordinates . Let us translate the triangle so is at the origin. Then . Using the fact that the slope of a line is equal to the tangent of the angle formed by the line and the positive x- axis, the value we now need to find is simply .
Using , and basic trigonometric identities, this simplifies to , so the answer is
See also
2005 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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