Chords and Their Distance From the Center

A chord is a segment connecting two points on a circle. The closer it sits to the center, the longer it is, because the longest possible chord passes right through the center as a diameter. Slide the chord away from the center and it has to "cut across" a smaller slice of the circle, so it shrinks. Right at the edge of the circle, the chord shrinks to a single point and has length zero.

The key move is dropping a perpendicular segment from the center straight down to the chord. That perpendicular always hits the chord at its exact midpoint, splitting it into two equal halves. This isn't a coincidence: the center, together with the two endpoints of the chord, forms an isosceles triangle, since both endpoints sit on the circle at distance r from the center. Drop a height from the top vertex of an isosceles triangle and it always bisects the base. That gives a right triangle with the radius r as the hypotenuse, the distance from center to chord as one leg, and half the chord as the other leg. From there it's the Pythagorean theorem.

This is the same toolkit used for central and inscribed angles and for tangent lines: circles keep producing right triangles once you draw the correct radius. Once you can set up this triangle, finding arc length or sector area from a chord, or working out where a tangent line touches, all use the same habit of connecting center to point with a radius or perpendicular.