Definition: A chord is a segment whose endpoints both lie on a circle; the longest chord is a diameter, and a perpendicular from the center bisects any chord.
What is a chord?
Pick any two points on a circle and connect them. The segment is a chord. A chord that passes through the center is a diameter, so the diameter is the longest chord.
Which theorems matter?
- A radius perpendicular to a chord bisects it.
- Congruent chords are the same distance from the center, and they cut congruent arcs.
- If two chords cross inside the circle, the products of their parts are equal.
Example. A chord of length 16 is 6 cm from the center. The half-chord is 8, so the radius is the square root of 64 + 36 = 10. If two chords intersect and one is cut into parts of 4 and 9, while the other has a part of 6, then 4(9) = 6x, so the other part is 6.
What goes wrong?
Using the full chord length where the half is needed in the right triangle. Also, confusing a chord with a secant, which is a line that extends beyond the circle, or a tangent, which touches at one point.