Mathematics Question on Number of Tangents from a Point on a Circle

Question

In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to a circle with the two tangents TP and TQ Fig. 10.11

Answer 1

Solution

It is given that $\text {TP}$ and $\text {TQ}$ are tangents.
Therefore, radius drawn to these tangents will be perpendicular to the tangents.
Thus, $\text {OP ⊥ TP }$ and $\text {OQ ⊥ TQ}$
$∠OPT = 90º$
$∠OQT = 90º$
In quadrilateral $POQT$ ,
Sum of all interior angles $= 360º$
$∠OPT + ∠POQ +∠OQT + ∠PTQ = 360º$
$⇒ 90º + 110º + 90º + PTQ = 360º$
$⇒ PTQ = 70º$

Hence, the correct option is (B): $70º$