WAEC 2020 · Paper 2 · Q11

The diagram is a circle, centre $O$ , with radius $35\text{ cm}$ . The arc $MN$ subtends an angle of $120^\circ$ at the centre. Find, correct to one decimal place, the: $\left[\text{Take }\pi = \frac{22}{7}\right]$

The paper marks this diagram “not drawn to scale”.

(a)
perimeter of the minor sector $MON$ ;
(b)
length of the chord $MN$ ;
(c)
area of the minor segment cut off by the chord $MN$ .

Worked solution (try it first)

(a)

Arc length $= \frac{120}{360} \times 2\pi r$
$= \frac13 \times 2 \times \frac{22}{7} \times 35$
$= \frac{220}{3}$
$\approx 73.33$ cm.
The sector's perimeter is the arc plus two radii: $73.33 + 70 = 143.3$ cm.

(b)

The perpendicular from $O$ bisects the chord and the $120^\circ$ angle.
Half the chord is $35\sin 60^\circ \approx 30.31$ cm, so $|MN| = 2 \times 30.31 \approx 60.6$ cm.

(c)

Area of the sector $= \frac13 \times \frac{22}{7} \times 35^2$
$= \frac{3850}{3}$
$\approx 1283.33\text{ cm}^2$ .
Area of triangle $MON = \frac12 \times 35 \times 35 \times \sin 120^\circ$
$\approx 530.44\text{ cm}^2$ .
Minor segment $=$ sector $-$ triangle $\approx 1283.33 - 530.44$
$= 752.9\text{ cm}^2$ .

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