Circles

Answer

Step 1: Use circumference formula C = 2πr C = 2π(7) = 14π cm ≈ 44 cm Step 2: Use area formula A = πr² A = π(7)² = 49π cm² ≈ 154 cm² Answer: Circumference = 14π cm, Area = 49π cm²

Answer

Step 1: Draw perpendicular OM from center O to chord AB Step 2: OM bisects AB, so AM = MB = 4 cm Step 3: In right triangle OAM: OA² = OM² + AM² OA² = 3² + 4² = 9 + 16 = 25 Step 4: OA = √25 = 5 cm Ans

Answer

Step 1: Given AB and CD are equal chords Step 2: Equal chords subtend equal angles at center (Theorem) Step 3: Since ∠AOB = ∠COD = 60° and both are equal chords Step 4: Therefore AB = CD Answer: CD =

Answer

Step 1: Use theorem - angle at center = 2 × angle at any point on circle Step 2: Let angle at center = θ Step 3: θ = 2 × 35° Step 4: θ = 70° Answer: Angle at center = 70°

Answer

Step 1: AB is diameter, so arc AB is a semicircle Step 2: Angle subtended by semicircle at center = 180° Step 3: Using theorem: angle at center = 2 × angle on circle Step 4: 180° = 2 × ∠APB Step 5: ∠A

Answer

Step 1: Chords equidistant from center are equal in length (Theorem) Step 2: Given: distance from O to PQ = distance from O to RS Step 3: Therefore PQ = RS Step 4: Since PQ = 8 cm Answer: RS = 8 cm

Answer

Step 1: In cyclic quadrilateral, opposite angles are supplementary Step 2: ∠A + ∠C = 180° Step 3: 75° + ∠C = 180° Step 4: ∠C = 180° - 75° = 105° Answer: ∠C = 105°

Answer

Step 1: Check if triangle is right-angled: 6² + 8² = 36 + 64 = 100 = 10² Step 2: Yes, it's right-angled at B Step 3: For right triangle, circumradius R = hypotenuse/2 Step 4: R = AC/2 = 10/2 = 5 cm A