In the diagram, is a right angle. Point is on , and bisects . Points and are on and , respectively, so that and . Given that and , find the integer closest to the area of quadrilateral .

Source: NCTM Mathematics Teacher

**SOLUTION**

*Angle Bisector Theorem
*The angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

*Divided Triangle Proposition*

When a triangle is divided into two smaller triangles by a segment from one of its vertices to the opposite side (whether or not this segment is an angle bisector), the ratio of the areas of the two smaller triangles is equal to the ratio of the triangles’ bases.

By the Pythagorean theorem

**Calculate the area of **

is the angle bisector of of . By the Angle Bisector theorem

By the Divided Triangle proposition

**Calculate the area of**

is the angle bisector of of . By the Angle Bisector theorem

By the Divided Triangle proposition

is a segment from vertex to point on opposite side in

is a segment from vertex to point on opposite side in

Finally

Integer closest to is .

**Answer**: square units