Circumference from the power of an interior point

Text

In a circle two chords meet at a point $P$ whose distance from the center is $6$ cm. The product of the two segments into which each chord is divided by $P$ equals $45$ cm$^2$. Find the radius and the length of the circle.

Solution

By the power of an interior point, if $R$ is the radius and $d=\overline{OP}$: $PA\cdot PB=R^2-d^2\ \Longrightarrow\ 45=R^2-36\ \Longrightarrow\ R^2=81\ \Longrightarrow\ R=9\ \text{cm}.$ Length of the circle: $C=2\pi R=18\pi\approx56.55\ \text{cm}.$ $\boxed{R=9\ \text{cm};\ \ C=18\pi\approx56.55\ \text{cm}}$