Question

Find the area of the shaded segment of circle O to the
nearest tenth of a square unit. The radius of the circle is 6 units. The perpendicular from O to the chord joining
point A to point B measures 4 units.
(a) 4.8 sq. units
(b) 12.4 sq. units
(C) 30.3 sq. units
(d) 48.2 sq. units
​

Answer 1

The radius of the circle is 6 units. The perpendicular from O to the chord joining point A to point B measures 4 units.

To find : The area of the shaded region of circle O to the nearest tenth of a square unit.

solution : here AP = √{6² - 4²} = √{20} = 2√5 unit

now ∠AOP = tan¯¹[AP/OP] = tan¯¹[2√5/4] = tan¯¹[√5/2] = 48.19°

so angle AOB = 2 × ∠AOP = 48.19 × 2 = 96.38°

now area of sector AOB = ∠AOB/360° × πr²

= 96.38/360 × (3.14) × (6)²

= 30.26 sq unit

and area of triangle AOB = 1/2 × height × base

= 1/2 × OP × AB

= 1/2 × 4 × 4√5

= 8√5 sq unit = 17.88 sq unit

so area of shaded region = area of sector AOB - area of triangle AOB

= 30.26 - 17.88

= 12.38 ≈ 12.4 sq unit

Therefore option (b) is correct choice.