JAMB 2017 · UTME (set 2) · Q48

Circle geometry
Trigonometric ratios

In the diagram, $MN$ is a chord of a circle $KMN$ with centre $O$ and radius $10\text{ cm}$ . If $\angle MON = 140^\circ$ , find, correct to the nearest cm, the length of the chord $MN$ .

Worked solution (try it first)

$OM = ON = 10$ cm.
The perpendicular from $O$ to $MN$ bisects the chord and the $140^\circ$ angle, giving two right-angled triangles with a $70^\circ$ angle at $O$ .
In one of them, half the chord is opposite the $70^\circ$ : $\frac{MN}{2} = 10\sin70^\circ = 9.397$ cm.
So $MN = 2 \times 9.397 = 18.79$ cm, which is 19 cm to the nearest cm, option A.

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