2015 AMC 10A Problems/Problem 19
The isosceles right triangle has right angle at and area . The rays trisecting intersect at and . What is the area of ?
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 .