Finding the perpendicular distance from a chord to the circumference

Question

Given a circle of circumference 60 and a chord of length 12, what is the perpendicular distance from the chord to the circumference?

What I don't understand about this the distance from the chord to the circumference means geometrically. Wouldn't that vary based on where on the chord you are?

(This question is apparently from BM 85194 (A Babylonian tablet).)

Answer 1

Let $O$ be the center of the circle, $C$ be the center of the chord, and $A$ be one endpoint of the chord.

Then $OCA$ is a right angle so triangle $OCA$ is a right triangle.

Therefore $OC^2+CA^2 = OA^2$, so $OC = \sqrt{OA^2-CA^2}$.

In this case, $OA = 60/(2\pi) =30/\pi$ and $AC = 12/2 = 6$ so $OC = \sqrt{(30/\pi)^2-6^2}$.

(added later)

So the distance from the center of the chord to the circle is the radius minus the distance from the center of the circle to the chord or, in this case, $30/\pi-\sqrt{(30/\pi)^2-6^2}$.