2015 AMC 10A Problems/Problem 19
Problem
The isosceles right triangle has right angle at
and area
. The rays trisecting
intersect
at
and
. What is the area of
?
Solution
Let and
be the points at which the angle trisectors intersect
.
can be split into a
right triangle and a
right triangle by dropping a perpendicular from
to side
. Let
be where that perpendicular intersects
.
Because the side lengths of a right triangle are in ratio
,
.
Because the side lengths of a right triangle are in ratio
and
+
,
.
Setting the two equations for equal together,
.
Solving gives .
The area of .
is congruent to
, so their areas are equal.
A triangle's area can be written as the sum of the figures that make it up, so .
.
Solving gives , so the answer is
.