WAEC 2020 · Paper 2 · Q4

(a)
In the diagram, $R$ , $S$ , $T$ , $U$ are points on a circle and $QRS$ is a straight line. $\angle STU = 124^\circ$ and $\angle QUR = 31^\circ$ . Find $\angle RQU$ .
(b)
The ratio of the length of an arc of a circle to the circumference of the circle is $3 : 7$ . If the diameter of the circle is $14\text{ cm}$ , calculate, correct to three significant figures, the: (i) perimeter of the minor sector; (ii) area of the minor sector. $\left[\text{Take }\pi = \frac{22}{7}\right]$
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

$RSTU$ is a cyclic quadrilateral, so opposite angles add up to $180^\circ$ : $\angle URS = 180^\circ - 124^\circ$
$= 56^\circ$ .
$QRS$ is a straight line, so $\angle URQ = 180^\circ - 56^\circ$
$= 124^\circ$ .
In triangle $QUR$ : $\angle RQU = 180^\circ - 124^\circ - 31^\circ$
$= 25^\circ$ .

The radius is 7 cm.
The arc is $\frac37$ of the circumference: $\frac37 \times 2 \times \frac{22}{7} \times 7 \approx 18.86$ cm.
Perimeter of the sector $= 18.86 + 7 + 7 \approx 32.9$ cm.

The sector is the same fraction $\frac37$ of the circle's area: $\frac37 \times \frac{22}{7} \times 7^2 = 66.0\text{ cm}^2$ .