1986 AIME Problems/Problem 15
Problem
Let triangle be a right triangle in the xy-plane with a right angle at . Given that the length of the hypotenuse is , and that the medians through and lie along the lines and respectively, find the area of triangle .
Solution
Translate so the medians are , and , then model the points and . is the centroid, and is the average of the vertices, so
so
(1)
AC and BC are perpendicular, so the product of their slopes is -1, giving
(2)
Combining (1) and (2), we get
Using the determinant product for area of a triangle (this simplifies nicely, add columns 1 and 2, add rows 2 and 3), the area is , so we get the answer to be .
See also
1986 AIME (Problems • Answer Key • Resources) | ||
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