Circle Relationships and Theorems
Students analyze relationships among central angles, inscribed angles, radii, chords, and tangents to solve problems involving circles.
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Parts of a Circle
A circle is the set of all points in a plane that are the same distance from a fixed center. A radius connects the center to a point on the circle. A diameter is a chord that passes through the center, so its length is twice the radius. A chord connects any two points on the circle, while an arc is part of the circle between two points. A secant crosses the circle at two points, but a tangent touches it at exactly one point. For example, suppose circle O has radius OE = 5 units. Any other radius also measures 5 units, and diameter AB measures 10 units. Chord CD may be shorter than the diameter because it does not pass through O.
Central and Inscribed Angles
A central angle has its vertex at the center of a circle, and its measure equals the measure of its intercepted arc. An inscribed angle has its vertex on the circle, and its sides are chords. Its measure is half the measure of its intercepted arc. Therefore, a central angle intercepting the same arc as an inscribed angle has twice the measure of the inscribed angle. For example, if central angle AOB measures 80°, then minor arc AB also measures 80°. An inscribed angle ACB that intercepts arc AB measures 40°. Inscribed angles intercepting the same arc are congruent. A useful special case occurs when an inscribed angle intercepts a semicircle: because a semicircle measures 180°, the inscribed angle measures 90°.
Chords and Radii
Chords have important relationships with the center and radii of a circle. A radius or diameter perpendicular to a chord bisects the chord, dividing it into two congruent segments. Conversely, a line from the center that bisects a chord is perpendicular to that chord. Congruent chords in the same circle are the same distance from the center and intercept congruent arcs. For example, circle O has radius 13 units, and chord AB is 5 units from O. If OM is perpendicular to AB, then M is the midpoint of AB. Triangle OMA is a right triangle, so AM equals the square root of 13² minus 5², or 12 units. Because AM = MB, the entire chord AB measures 24 units.
Tangents and Perpendicular Radii
A tangent line touches a circle at exactly one point, called the point of tangency. The radius drawn to that point is perpendicular to the tangent. This relationship creates a right angle that can be used in proofs and calculations. Two tangent segments drawn from the same external point are congruent. Also, an angle formed by two tangents is a circumscribed angle, and its measure equals 180° minus the measure of the intercepted minor arc. For example, tangents PA and PB touch circle O at A and B. If minor arc AB measures 110°, then angle APB measures 180° − 110°, or 70°. In addition, PA = PB, and radii OA and OB form right angles with the tangent segments.
Applying Circle Theorems
Circle problems often require several theorems to be used together. Begin by identifying each angle’s vertex and sides. A vertex at the center indicates a central angle, a vertex on the circle indicates an inscribed angle, and a vertex outside the circle may indicate a circumscribed angle. Then identify the intercepted arc. For example, suppose minor arc AB measures 124°. Central angle AOB also measures 124°, while inscribed angle ACB measures half as much, or 62°. If angle ACB is labeled 3x + 2 degrees, then 3x + 2 = 62, so x = 20. If tangents PA and PB meet at external point P, circumscribed angle APB measures 180° − 124° = 56°. The radii OA and OB are also perpendicular to the tangents.