WAEC 2012 · Paper 2 · Q5

In the diagram, $O$ is the centre of the circle and $XY$ is a chord. If the radius is $5\text{ cm}$ and $|XY| = 6\text{ cm}$ , calculate, correct to 2 decimal places, the: $\left[\text{Take }\pi = \frac{22}{7}\right]$

(a)
angle which $XY$ subtends at the centre $O$ ;
(b)
area of the shaded minor segment.

Worked solution (try it first)

(a)

Join $O$ to $X$ and $Y$ .
The triangle $OXY$ is isosceles ( $OX = OY = 5$ ), so the perpendicular from $O$ bisects the chord: each half is 3 cm, and it cuts the angle $\theta$ at $O$ in half.
In one right-angled half, $\sin\frac\theta2 = \frac35 = 0.6$ .
So $\frac\theta2 \approx 36.87^\circ$ and $\theta \approx 73.74^\circ$ .

(b)

Area of the sector $OXY = \frac{\theta}{360} \times \pi r^2$
$= \frac{73.74}{360} \times \frac{22}{7} \times 25$
$\approx 16.09\text{ cm}^2$ .
The perpendicular from $O$ to $XY$ is $\sqrt{5^2 - 3^2} = 4$ cm, so the area of triangle $OXY = \frac12 \times 6 \times 4$
$= 12\text{ cm}^2$ .
Minor segment $=$ sector $-$ triangle $\approx 16.09 - 12 = 4.09\text{ cm}^2$ .

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