WAEC 2013 · Paper 2 · Q5✱

(a)
The pie chart shows the result of a survey conducted on the maintenance of five small and medium scale industries in a state. If the cost of maintenance of the plywood industry was ₦34,500.00, calculate the cost of maintenance of the textile industry.
(b)
In the diagram, $TYZ$ and $UWZ$ are tangents to the circle $WXY$ , $\angle XYZ = 100^\circ$ , $\angle YZW = 42^\circ$ and $\angle UWX = m$ . Find the value of $m$ .

Worked solution (try it first)

The angles at the centre add up to $360^\circ$ : $x + x + 82^\circ + 68^\circ + 72^\circ = 360^\circ$ .
So $2x + 222^\circ = 360^\circ$ , which gives $2x = 138^\circ$ and $x = 69^\circ$ .
The plywood sector is $69^\circ$ .
$69^\circ$ stands for ₦34,500, so the whole $360^\circ$ stands for $\frac{360}{69} \times 34\,500 = 180\,000$ .
The total cost is ₦180,000.
The textile sector is $68^\circ$ : $\frac{68}{360} \times 180\,000 = 34\,000$ .
The maintenance of the textile industry cost ₦34,000.00.

Tangents from the same point are equal, so $|ZY| = |ZW|$ and triangle $ZYW$ is isosceles with $\angle ZYW = \angle ZWY$ .
Angles in triangle $ZYW$ : $\angle ZYW = \frac{180^\circ - 42^\circ}{2}$
$= 69^\circ$ .
Alternate segment theorem: the angle between the tangent $WU$ and the chord $WX$ equals the angle in the alternate segment, so $\angle XYW = \angle UWX = m$ .
At $Y$ , $\angle XYZ = \angle XYW + \angle WYZ$ , so $100^\circ = m + 69^\circ$ .
So $m = 31^\circ$ .