2009 AMC 12A Problems/Problem 14
Problem
A triangle has vertices ,
, and
, and the line
divides the triangle into two triangles of equal area. What is the sum of all possible values of
?
Solution
Let's label the three points as ,
, and
.
Clearly, whenever the line intersects the inside of the triangle, it will intersect the side
. Let
be the point of intersection.
The triangles and
have the same height, which is the distance between the point
and the line
.
Hence they have equal areas if and only if
is the midpoint of
.
The midpoint of the segment has coordinates
. This point lies on the line
if and only if
. This simplifies to
. This is a quadratic equation with roots
and
. Both roots represent valid solutions, and their sum is
.
For illustration, below are pictures of the situation for ,
,
, and
.
See Also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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All AMC 12 Problems and Solutions |