WAEC 2024 · Paper 2 · Q13

(a)
In the diagram, $O$ is the centre of the circle, $PQ$ is a tangent to the circle at $T$ and $ABC$ is a straight line. $TC$ bisects $\angle BTQ$ , $\angle BAT = 44^\circ$ and $\angle PTA = 60^\circ$ . Find $\angle ACT$ .
(b)
The circumference of the base of a cylindrical tank is $11\text{ m}$ . The height of the tank is $3\text{ m}$ more than 6 times the base radius. Calculate the: (i) radius; (ii) height; (iii) volume of the tank. $\left[\text{Take } \pi = \frac{22}{7}\right]$
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

By the alternate segment theorem, the angle between the tangent $TQ$ and the chord $TB$ equals the angle in the alternate segment: $\angle BTQ = \angle BAT = 44^\circ$ .
$TC$ bisects it, so $\angle BTC = 22^\circ$ .
Also by the alternate segment theorem, $\angle ABT = \angle PTA = 60^\circ$ .
$ABC$ is a straight line, so $\angle ABT$ is an exterior angle of triangle $BTC$ , and it equals the sum of the two opposite interior angles: $60^\circ = 22^\circ + \angle ACT$ .
So $\angle ACT = 38^\circ$ .