### Instruction: A circle has a radius of three inches. The distance from the center of the circle to chord $CD$ is two inches. How many inches long is chord $CD$? Express your answer in simplest radical form.### Response: I need to find the length of chord $CD$, so I should draw a picture of the situation.
I draw a circle with center $O$ and radius $3$, and I mark chord $CD$ and the distance from $O$ to $CD$, which is $2$.
I see that I can form a right triangle by drawing a line segment from $O$ to the midpoint of $CD$, which I call $M$.
This triangle has a right angle at $M$, a hypotenuse of $3$, and a leg of $2$.
I can use the Pythagorean theorem to find the length of the other leg, which is $OM$.
$OM^2 + 2^2 = 3^2$
$OM^2 = 9 - 4$
$OM^2 = 5$
$OM = \sqrt{5}$
Now I have the length of $OM$, which is half of the length of $CD$.
To find the length of $CD$, I just need to multiply $OM$ by $2$.
$CD = 2 \cdot OM$
$CD = 2 \cdot \sqrt{5}$