Theorem 12-5
Relationship between chords and central angles
In a circle, a chord and its corresponding central angle have the following relationship:
1. When two chords in a circle intersect, the product of their segments is equal.
2. The measure of an inscribed angle is half the measure of its corresponding central angle. Conversely, the measure of a central angle is twice the measure of its corresponding inscribed angle.
3. Two chords in a circle are congruent if and only if they are equidistant from the center of the circle.
4. If two chords in a circle are equidistant from the center of the circle, then they are congruent.
5. The perpendicular bisector of a chord in a circle passes through the center of the circle. Conversely, any line passing through the center of a circle and perpendicular to a chord bisects the chord.
Theorem 12-5 states that for any given central angle in a circle, the measure of the corresponding chord is equal to twice the radius multiplied by the sine of half the angle.
In other words, let's say we have a circle with center O and radius r. Suppose we have a central angle with measure θ. Let A and B be the endpoints of the chord formed by this angle.
According to Theorem 12-5, the length of chord AB is given by AB = 2rsin(θ/2).
This theorem establishes a relationship between the measure of a central angle in a circle and the length of the chord formed by that angle.