2005 AIME I Problems/Problem 10
Problem
Triangle lies in the Cartesian Plane and has an area of 70. The coordinates of
and
are
and
respectively, and the coordinates of
are
The line containing the median to side
has slope
Find the largest possible value of
Solution
The midpoint of line segment
is
. Let
be the point
, which lies along the line through
of slope
. The area of triangle
can be computed in a number of ways (one possibility: extend
until it hits the line
, and subtract one triangle from another), and each such calculation gives an area of 14. This is
of our needed area, so we simply need the point
to be 5 times as far from
as
is. Thus
, and the sum of coordinates will be larger if we take the positive value, so
and the answer is
.