Circles

Answer

Step 1: Apply Theorem 9.1 - Equal chords subtend equal angles at center. Step 2: Since ∠AOB = ∠COD = 60° (given) Step 3: By the theorem, chords are equal when angles are equal Step 4: Therefore, CD =

Answer

Step 1: Draw perpendicular OM from center O to chord PQ, OM = 12 cm Step 2: OM bisects PQ (perpendicular from center bisects chord) Step 3: In right triangle OMP: OP² = OM² + MP² Step 4: 13² = 12² + M

Answer

Step 1: Apply Theorem 9.7 - Angle at center = 2 × Angle at circumference Step 2: Let angle at circumference = θ Step 3: Angle at center = 2θ = 80° Step 4: Therefore, θ = 80°/2 = 40° Answer: The arc su

Answer

Step 1: Apply property of cyclic quadrilateral - opposite angles are supplementary Step 2: ∠P + ∠R = 180° and ∠Q + ∠S = 180° Step 3: Find ∠R: 70° + ∠R = 180° Step 4: ∠R = 180° - 70° = 110° Step 5: Fin

Answer

Step 1: In a semicircle, the arc is exactly half the circle Step 2: The angle subtended by diameter at center = 180° Step 3: Apply angle theorem: Angle at circumference = (1/2) × Angle at center Step

Answer

Step 1: When two chords intersect inside a circle, vertically opposite angles are equal Step 2: ∠APD and ∠BPC are vertically opposite angles Step 3: Therefore, ∠BPC = ∠APD Step 4: ∠BPC = 75° Alternati

Answer

Step 1: Let O be center, AB be chord, OM be perpendicular from O to AB Step 2: OM = 6 cm (given), AB = 16 cm Step 3: OM bisects AB, so AM = MB = 16/2 = 8 cm Step 4: In right triangle OMA: OA² = OM² +

Answer

Use this theorem when: 1. You have two equal chords and need to find their distances from center 2. You know distances are equal and need to prove chords are equal Example: Chords AB = CD = 10 cm in a