Question

In the accompanying diagram of circle O, diameter AB is perpendicular to chordCD and intersects CDat E, CD = 12 and AB = 20.What is the length of OE?

Answer 1

`OE=8`

Explanation:

A diameter that is perpendicular to a chord divides the chord into two equal parts.

`CE=DE`

Therefore, if CD=12 and AB=20

The perpendicular bisector of a chord passes through the center of the circle

`\begin{gathered} CD=12 \\ CE=6=DE \end{gathered}`

Since AB is the diameter of the circle and O is the center of the circle, AO=10, and OB=10. CO is a radius, then CO=10.

We can use the Pythagorean theorem to find the measure for OE:

`\begin{gathered} OE^2+6^2=10^2 \\ OE=\sqrt[]{10^2-6^2} \\ OE=\sqrt[]{100-36} \\ OE=\sqrt[]{64}=8 \end{gathered}`