Revise: Circles
Central and inscribed angles, the angle in a semicircle, cyclic quadrilaterals, and the properties of chords — equal chords, the perpendicular bisector, and intersecting chords.
Inscribed angle = (1/2) × central angle (same arc).
Central 80° → inscribed 40°.
An angle inscribed in a semicircle is always 90°.
∠CAB=35° → ∠CBA=55° (since ∠ACB=90°).
Opposite angles of a cyclic quadrilateral sum to 180°.
∠A=105° → ∠C=75°.
A perpendicular from the centre bisects the chord.
r=13, distance=5 → chord=24.
Inscribed angle = half the central angle subtending the same arc.
Inscribed 35° → central 70°.
A diameter's arc is 180°, so any inscribed angle on it is 90°.
∠ABC=62° → ∠BAC=28°.
Opposite angles sum to 180° — their arcs together make the full circle.
∠Q=98° → ∠S=82°.
Equal chords ⇔ equal distance from centre ⇔ equal arcs.
Equal distance from centre → equal chord length.
OM ⊥ AB ⇒ AM = MB; sets up radius² = distance² + half-chord².
r=10, distance=6 → chord=16.
Chords crossing inside a circle: PA×PC = PB×PD.
PA=5,PC=9,PD=15 → PB=3.