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 |