WAEC Mathematics 2012 Theory question 105

Theory question 105

Mensuration of Plane Shapes

In the diagram, O is the centre of the circle and XY is a chord. If the radius is 5 cm and |XY| = 6 cm, calculate, correct to 2 decimal places, the: (a) angle which XY subtends at the centre O; (b) area of the shaded portion.

Show the model answer

Model answer

(a) Half the chord is 3 cm: sin(½θ) = 3/5 = 0.6, so ½θ = 36.87° and θ = 73.74°.
(b) Sector = 73.74/360 × 22/7 × 5² ≈ 16.09 cm².
Triangle OXY = ½ × 6 × 4 = 12 cm² (height from O = √(25 − 9) = 4).
Shaded segment ≈ 16.09 − 12 = 4.09 cm².

Explanation

The shaded minor segment is the sector minus the triangle.