2013 AMC 12B Problems/Problem 8
Problem
Line has equation and goes through . Line has equation and meets line at point . Line has positive slope, goes through point , and meets at point . The area of is . What is the slope of ?
Solution 1
Line has the equation when rearranged. Substituting for , we find that line will meet this line at point , which is point . We call the base and the altitude from to the line connecting and , , the height. The altitude has length , and the area of . Since , . Because has positive slope, it will meet to the right of , and the point to the right of is . passes through and , and thus has slope .
Solution 2 - Shoelace Theorem
We know lines and intersect at , so we can solve for that point: Because we have:
Thus we have .
We know that the area of the triangle is , so by Shoelace Theorem we have:
Thus we have two options:
or
Now we must just find a point that satisfies is positive.
Doing some guess-and-check yields, from the second equation:
so a valid point here is . When calculated, the slope of in this situation yields .
Video Solution
~Punxsutawney Phil or sugar_rush
See also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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