2008 AMC 12B Problems/Problem 17
Problem
Let , and be three distinct points on the graph of such that line is parallel to the -axis and is a right triangle with area . What is the sum of the digits of the -coordinate of ?
Solution
Supposing , is perpendicular to and, it follows, to the -axis, making a segment of the line . But that would mean that the coordinates of are , contradicting the given that points and are distinct. So is not . By a similar logic, neither is .
This means that and is perpendicular to . Let C be the point . So the slope of is the negative reciprocal of the slope of , yielding .
Because is the length of the altitude of triangle from , and is the length of , the area of . Since , . Substituting, , whose digits sum to .
See Also
2008 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.