11. A quadrilateral contains an inscribed circle (i.e. a circle tangent to the four sides of the quadrilateral). The ratio of the perimeter of the quadrilateral to that of the circle is $4: 3$. The ratio of the area of the quadrilateral to that of the circle, expressed in its lowest terms, is $a: b$. What is the value of $100 a+49 b$ ?
SolUtion
547

Let the radius of the circle be 3 . Then its circumference is $6 \pi$ and the quadrilateral's perimeter is $8 \pi$. The circle's area is $9 \pi$. We can determine the quadrilateral's area by dividing it into four triangles, as shown in the example. Each triangle has area $\frac{1}{2} \times 3 \times s$, where $s$ is the side shared by the triangle and the quadrilateral.
By summing the triangles' areas we find the quadrilateral's area, as $\frac{1}{2} \times 3 \times$ (the quadrilateral's perimeter), this is $\frac{1}{2} \times 3 \times 8 \pi=12 \pi$. The required ratio of areas is $12 \pi: 9 \pi=4: 3$. Therefore $a=4, b=3$, and $100 a+49 b=547$.