JAMB 1983 · UME · Q49

In the figure, $PT$ is a tangent to the circle with centre $O$ . If $\angle PQT = 30^\circ$ , find the value of $\angle PTO$ .

Worked solution (try it first)

A tangent is perpendicular to the radius at the point of contact, so $\angle OPT = 90^\circ$ and $\angle QPT = x + 90^\circ$ .
At $T$ the two marked angles make $\angle QTP = x + 2x = 3x$ .
Angles in triangle $PQT$ add up to $180^\circ$ : $30^\circ + (x + 90^\circ) + 3x = 180^\circ$ , so $4x = 60^\circ$ and $x = 15^\circ$ .
So $\angle PTO = 2x = 30^\circ$ , option A.