Circle tricks that make arc problems click

Inscribed angle = ½ intercepted arc. Central angle = arc.

Inscribed Angle Theorem — Vertex on circle: inscribed angle = ½ arc. Vertex at center: central angle = arc. Inscribed angles intercepting the same arc are equal to each other.

Tangent is perpendicular to radius at point of tangency

Tangent-Radius Relationship — This right angle enables Pythagorean theorem solutions on tangent problems. Two tangent segments from an external point are always equal in length.

Sector area = (θ/360) × πr². Arc length = (θ/360) × 2πr.

Sector and Arc Formulas — Central angle θ out of 360° total. Area and arc length are both proportional fractions of the full circle. For radians: area = ½r²θ, arc length = rθ.

Intersecting chords: segment₁ × segment₂ = segment₃ × segment₄

Chord Product Theorem — Chord AB and chord CD intersect at point P: AP × PB = CP × PD. For secants from external point: (whole) × (external) = (whole) × (external).

Power of a Point: for any point and circle, the product is constant regardless of chord direction

Power of a Point — Interior: two chords → segment products equal. Exterior: two secants → (whole)(external) = (whole)(external). One tangent, one secant → tangent² = (whole secant)(external secant). All follow from the same power.

Chord: segment with both endpoints on circle. Diameter: chord through center. Longest chord = diameter.

Circle Vocabulary — Radius: center to any point on circle. Diameter: twice the radius, chord through center. Chord: any segment connecting two points on the circle. Secant: line intersecting circle at two points. Tangent: line touching circle at exactly one point (point of tangency). Arc: portion of the circle between two points.

Congruent chords are equidistant from center. Chord perpendicular from center bisects the chord.

Chord-Distance Theorems — A perpendicular from the center to a chord bisects the chord. Converse: the segment from center to the midpoint of a chord is perpendicular to the chord. Congruent chords are equidistant from the center. Chords equidistant from center are congruent.

Angles formed by two secants from external point = ½(difference of intercepted arcs)

Inside circle½ × sum of intercepted arcs

On circle (inscribed)½ × intercepted arc

Outside circle½ × difference of intercepted arcs

Tangent-tangent angle: two tangents from external point → angle = ½(major arc - minor arc)

Tangent Lines from External Point — Two tangent segments from an external point are equal in length. The angle between them = ½(major arc - minor arc). The line from the external point to the center bisects the angle between the tangents. The two radii to the tangent points are perpendicular to the tangents.

Radian measure: arc length = rθ. Full circle = 2π radians = 360°. 1 radian ≈ 57.3°.

Radian Measure — 1 radian: the angle subtended by an arc equal in length to the radius. 2π radians = 360° (full circle). Convert degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Arc length = rθ (r = radius, θ in radians). Area of sector = ½r²θ.

Concentric circles: same center, different radii. Congruent circles: same radius (may be different centers).

Circle Relationships — Concentric circles: share the same center but have different radii — like a bullseye. The area between two concentric circles = annulus (ring). Congruent circles: same radius, different centers — all points in one can be mapped to the other by translation. Equal radii ↔ congruent circles.

Circle equation in general form: x² + y² + Dx + Ey + F = 0 → complete the square to find center and radius

Circle Equation — General Form — Standard form: (x-h)² + (y-k)² = r². General form: x² + y² + Dx + Ey + F = 0. Convert: group x and y terms, complete the square for each. x² + Dx → (x + D/2)² - (D/2)². If result is positive: circle. Zero: single point. Negative: no real circle.