Question 7 of 7intermediate🔧 ApplyNumerical4 marks

A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use π = 3.14 and 3\sqrt{3} = 1.73)

Correct Answer

88.44 cm²

Exercise: EXERCISE 11.1 | Q: 7 | (Chapter: Page 5)
For More Understanding

Explanation

This question tests the concept of area of a segment of a circle. The segment area is found by subtracting the area of the triangle formed by the chord and two radii from the area of the sector. The context provides the formula: Area of segment = Area of sector – Area of ΔOAB. Example 2 in the context demonstrates the exact same problem structure with a 120° angle, showing how to find the triangle area using trigonometry (sin 60° = 3\sqrt{3}/2).

Solution Steps

  1. Step 1: Find the area of the sector OAPB

  2. Area of sector = (θ/360) × πr² = (120/360) × 3.14×123.14 \times 12 × 12 = (1/3) × 452.16 = 150.72 cm²

  3. Step 2: Find the area of ΔOAB

  4. Draw OM ⊥ AB. Since OA = OB, M is the mid-point of AB and ∠AOM = 60°

  5. Area of ΔOAB = (1/2) × OA × OB × sin(120°) = (1/2) × 12×1212 \times 12 × (3\sqrt{3}/2) = 363\sqrt{3} = 36×1.7336 \times 1.73 = 62.28 cm²

  6. Step 3: Find the area of the segment

  7. Area of segment = Area of sector – Area of ΔOAB = 150.72 – 62.28 = 88.44 cm²

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