2006 SMT/Geometry Problems/Problem 2
Problem
Given square of side length
, with
on
and
in the interior of the square so that
and
, find the area of quadrilateral
.
Solution
Since ,
lies on the perpendicular bisector of
, which is the same as the perpendicular bisector of
. Therefore, since
is the foot of the perpendicular from
to
,
is the midpoint of
, and
.
Let be the foot of the perpendicular from
to
. Since
is a rectangle,
. Let
. We have
, and
. Therefore, from right triangle
,
, and solving this gives
.
Finally, is a trapezoid with bases of length
and
and height
, so its area is
.