Circles
Circles
The Circles domain tests angle and segment relationships in circles, arc and chord properties, tangents, and cyclic quadrilaterals. Regents questions often combine a diagram-free description with a short computation or a justification.
Core rules
- A central angle equals its intercepted arc. An inscribed angle is half its intercepted arc.
- An angle inscribed in a semicircle is a right angle. In a cyclic quadrilateral, opposite angles are supplementary.
- A tangent is perpendicular to the radius at the point of tangency. Tangent segments from the same external point are congruent.
- The angle formed by two chords intersecting inside a circle is half the sum of the intercepted arcs. The angle formed by two secants or tangents from an external point is half the difference of the intercepted arcs.
- Chord properties: a perpendicular from the center to a chord bisects the chord; congruent chords are equidistant from the center.
- Secant segments from an external point satisfy outside times whole equals outside times whole, often written PA times PB equals PC times PD.
Worked example
From external point P, secant PAB meets the circle at A then B, and secant PCD meets it at C then D. PA equals 4, AB equals 6, and PC equals 5. Find CD. PB equals PA plus AB, or 10. By the secant-segment theorem, PA times PB equals PC times PD, so 4 times 10 equals 5 times PD, giving PD equals 8. Since PD equals PC plus CD, 8 equals 5 plus CD, so CD equals 3.
Traps
Do not confuse arc measure in degrees with arc length in linear units. An inscribed angle uses the arc not containing the vertex. For an external angle, use the difference of arcs, not the sum. When using secant products, multiply the outside segment by the entire secant, not just the inside part. Do not assume a line is tangent unless a right angle or tangent statement is given.