# Ex.10.2 Q2 Circles Solution - NCERT Maths Class 10

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## Question

In the given figure, if $$TP$$ and $$TQ$$ are the two tangents to a circle with centre $$O$$ so that $$\angle {P O Q} = 110 ^ { \circ }$$, then $$\angle {P T Q}$$ is equal to

(A)   $$60^\circ$$

(B)   $$70^\circ$$

(C)   $$80^\circ$$

(D)   $$90^\circ$$ ## Text Solution

What is Known?

(i) $$TP$$ and $$TQ$$ are tangents to a circle with Centre $$O$$

(ii) $$\angle {P O Q}=110^{\circ}$$

What is Unknown?

$$\angle {PTQ}$$

Reasoning:

• Tangent at any point of a circle is  perpendicular to the radius through the point of contact.
• In the above figure $$OPTQ$$ is a quadrilateral and $$\angle {P}$$ and $$\angle {Q}$$  are $$90^{\circ}$$
• Sum of the angles of a quadrilateral is $$360^{\circ}$$

Steps:

$$\therefore \;$$In $$OPTQ,$$

\begin{align} \angle {Q} + \angle {P} + \angle {P O Q} + \angle {P T Q} &= 360 ^ { \circ }\\ 90 ^ { \circ } + 90 ^ { \circ } + 110 ^ { \circ } + \angle {P T Q} &= 360 ^ { \circ } \\ 290 ^ { \circ } + \angle {P T Q} &= 360 ^ { \circ } \\ \angle {P T Q} &= 360 ^ { \circ } - 290 ^ { \circ } \\ \angle {P T Q} &= 70 ^ { \circ } \end{align}

Hence the correct Option is B

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