In the given figure if tp and tq are the two tangents to a circle with centre o so that poq=120

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If TP and TQ are two tangents to a circle with center O such that ∠ POQ =110∘, then ∠ PTQ will be A. 70∘B. 80∘С. 90∘

${70}^{\circ }$

${80}^{\circ }$

${90}^{\circ }$

Solution

The correct option is A ${70}^{\circ }$

$\text{Given:}\angle POQ={110}^{\circ }$ TP and TQ are tangents. A tangent at any point of a circle is perpendicular to the radius at the point of contact.

$\Rightarrow \angle OQT=\angle OPT={90}^{\circ }$

We know that, sum of interior angles of a quadrilateral is 360 degrees.

$\angle TQO+\angle QOP+\angle OPT+\angle PTQ={360}^{\circ }$

${90}^{\circ }+{110}^{\circ }+{90}^{\circ }+\angle PTQ={360}^{\circ }\Rightarrow \angle PTQ={360}^{\circ }–{90}^{\circ }–{90}^{\circ }–{110}^{\circ }\Rightarrow \angle PTQ={70}^{\circ }\therefore \angle PTQ={70}^{\circ }$