WAEC 2012 · Paper 2 · Q3

(a)
In the diagram, $\overline{TU}$ is a tangent to the circle. $\angle RVU = 100^\circ$ and $\angle URS = 36^\circ$ . Calculate the value of angle $STU$ .
(b)
In triangle $XYZ$ , $|XY| = 5\text{ cm}$ , $|YZ| = 8\text{ cm}$ and $|XZ| = 6\text{ cm}$ . $P$ is a point on the side $XY$ such that $|XP| = 2\text{ cm}$ and the line through $P$ , parallel to $YZ$ , meets $XZ$ at $Q$ . Calculate $|QZ|$ .

Worked solution (try it first)

$RVUS$ is a cyclic quadrilateral, and $RST$ is a straight line.
The exterior angle of a cyclic quadrilateral equals the interior angle opposite it: $\angle UST = \angle RVU = 100^\circ$ .
$TU$ is a tangent at $U$ , so by the alternate segment theorem the angle between the tangent and the chord $US$ equals the angle in the other segment: $\angle SUT = \angle URS = 36^\circ$ .
In triangle $SUT$ : $\angle STU = 180^\circ - 100^\circ - 36^\circ$
$= 44^\circ$ .

Draw triangle $XYZ$ with $P$ on $XY$ , $|XP| = 2$ cm, and $PQ \parallel YZ$ meeting $XZ$ at $Q$ .
Triangles $XPQ$ and $XYZ$ have the same angles (corresponding angles), so they are similar, with $XQ$ matching $XZ$ and $XP$ matching $XY$ .
$\frac{|XQ|}{|XZ|} = \frac{|XP|}{|XY|}$ : $\frac{|XQ|}{6} = \frac25$ , so $|XQ| = 2.4$ cm.
Then $|QZ| = 6 - 2.4 = 3.6$ cm.