Find the radius of the circle

**Problem:** Circumcircle with center $P$ is shown below. A chord of the circle is positioned above the center. The perpendicular distance from $P$ to that chord is $4\ \text{cm}$, and half of the chord (from the midpoint to one end) is $3\ \text{cm}$ (the figure shows a point $O$ aligned with that end and the length from the end to $O$ is labeled $3\ \text{cm}$). What is the radius of the circle? **Answer:** Drop a perpendicular from the center $P$ to the chord; it bisects the chord. This forms a right triangle with legs $3\ \text{cm}$ and $4\ \text{cm}$ and hypotenuse equal to the radius. Hence the radius is \[ r=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\ \text{cm}. \] So the radius is **5 cm**.