WAEC 2017 · Paper 2 · Q9

(a)
$PQ$ is a tangent to a circle $RST$ at the point $S$ . $PRT$ is a straight line, $\angle TPS = 34^\circ$ and $\angle TSQ = 65^\circ$ . (i) Illustrate the information in a diagram. (ii) Find the value of: (I) $\angle RTS$ ; (II) $\angle SRP$ .
Separate values with commas, e.g. 3, −2
(b)
In the diagram, $XVY$ and $XWZ$ are straight lines cutting the circle $VYZW$ . $|VZ| = |YZ|$ , $\angle YXZ = 20^\circ$ and $\angle ZVY = 52^\circ$ . Calculate the size of $\angle WYZ$ .

Worked solution (try it first)

Draw the circle through $R$ , $S$ and $T$ .
Draw the tangent $PQ$ touching it at $S$ , and the straight line from $P$ through $R$ to $T$ .
Mark $\angle TPS = 34^\circ$ and $\angle TSQ = 65^\circ$ .

(I)** $\angle TSQ$ is an exterior angle of triangle $PST$ , so it equals the sum of the two opposite interior angles: $65^\circ = 34^\circ + \angle PTS$ .
So $\angle RTS = 31^\circ$ .
(II) By the alternate segment theorem, the angle between the tangent $SQ$ and the chord $ST$ equals the angle in the alternate segment: $\angle SRT = 65^\circ$ .
$P$ , $R$ , $T$ are on a straight line, so $\angle SRP = 180^\circ - 65^\circ$
$= 115^\circ$ .

$X$ , $V$ , $Y$ are on a straight line, so $\angle XVZ = 180^\circ - 52^\circ$
$= 128^\circ$ .
In triangle $XVZ$ : $\angle XZV = 180^\circ - 20^\circ - 128^\circ$
$= 32^\circ$ .
$|VZ| = |YZ|$ , so $\angle VYZ = \angle ZVY = 52^\circ$ .
$\angle VYW$ and $\angle VZW$ stand on the same arc $VW$ , so $\angle VYW = 32^\circ$ (angles in the same segment).
So $\angle WYZ = 52^\circ - 32^\circ$
$= 20^\circ$ .