2008 AMC 12B Problems/Problem 17
Let the coordinates of be and the coordinates of be . Since the line is parallel to the -axis, the coordinates of must be . Then the slope of line is . The slope of line is .
Supposing , is perpendicular to and, it follows, to the -axis, making a segment of the line x=m. 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 . 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 .