Circles

13 cm

Step 1: Let distances from centre to chords be x and x+7. Let chord of 24 cm be at distance x, and 10 cm chord at distance x+7. Step 2: For 24 cm chord: r² = x² + 12² = x² + 144. Step 3: For 10 cm chord: r² = (x+7)² + 5² = (x+7)² + 25. Step 4: Equating: x² + 144 = (x+7)² + 25, so x² + 144 = x² + 14x + 49 + 25, giving 14x = 70, so x = 5. Step 5: Therefore, r² = 5² + 12² = 25 + 144 = 169, so r = 13 cm.

616 cm²

Step 1: Given diameter = 28 cm, so radius = 28/2 = 14 cm. Step 2: Using the formula: Area = πr². Step 3: Substituting values: Area = (22/7) × 14². Step 4: Calculating: Area = (22/7) × 196 = 22 × 28 = 616. Step 5: Therefore, area = 616 cm².

176 cm²

Step 1: Area between circles = Area of outer circle - Area of inner circle. Step 2: Area of outer circle = π × 10² = (22/7) × 100 = 2200/7. Step 3: Area of inner circle = π × 6² = (22/7) × 36 = 792/7. Step 4: Area between = 2200/7 - 792/7 = (2200-792)/7 = 1408/7. Step 5: Therefore, area between = 1408/7 = 176 cm².

12√3 cm

Step 1: Given central angle = 120° and radius = 12 cm. Step 2: For a chord subtending angle θ at centre: chord length = 2r sin(θ/2). Step 3: Here, θ/2 = 120°/2 = 60°. Step 4: Chord length = 2 × 12 × sin(60°) = 24 × (√3/2) = 12√3. Step 5: Therefore, chord length = 12√3 cm ≈ 20.8 cm.