11. The diagram shows a circle with centre $O$ and a triangle $O P Q$. Side $P Q$ is a tangent to the circle. The area of the circle is equal to the area of the triangle.

What is the ratio of the length of $P Q$ to the circumference of the circle?
A $1: 1$
B $2: 3$
C $2: \pi$
D $3: 2$
E $\pi: 2$
Solution
A Suppose that the circle has radius $r$ and that $P Q$ has length $x$. The height of the triangle $O P Q$ is the length of the perpendicular from $O$ to $P Q$. Since $P Q$ is a tangent to the circle, this perpendicular is a radius of the circle and so has length $r$. Therefore the area of triangle $O P Q$ is $\frac{1}{2} x r$. The area of the circle is $\pi r^{2}$. Since these areas are equal $\frac{1}{2} x r=\pi r^{2}$, and hence $x=2 \pi r$. So the length of $P Q$ is the same as the circumference of the circle. So the ratio of their lengths is $1: 1$.