In Q.1 to 3, choose the correct option and give justification. 2. In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that , then is equal to (A) (B) (C) (D)

Understand the Question The tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, and . In quadrilateral , the sum of interior angles is . Since the two angles at the points of contact sum to , the central angle and the angle between the tangents are supplementary: . Step 1 · Find Angles at the Points of Contact Since the radius is perpendicular to the tangent at the point of contact: Step 2 · Calculate using Angle Sum Property In quadrilateral , the sum of all interior angles is : Substitute the known values: Answer > (B) Common Mistakes Direct Supplementary Shortcut: Forgetting that the angle subtended by the tangents () and the central angle () are directly supplementary: . Tangent-Radius Angle Error: Assuming the angle between the tangent and radius is something other than .

Circles | Exercise 10.2

Question 2

In Q.1 to 3, choose the correct option and give justification.

In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that $\angle \text{POQ} = 110^\circ$ , then $\angle \text{PTQ}$ is equal to (A) $60^\circ$ (B) $70^\circ$ (C) $80^\circ$ (D) $90^\circ$

Solution

Understand the Question

The tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, $\angle \text{OPT} = 90^\circ$ and $\angle \text{OQT} = 90^\circ$ .
In quadrilateral $\text{OPTQ}$ , the sum of interior angles is $360^\circ$ .
Since the two angles at the points of contact sum to $180^\circ$ , the central angle $\angle \text{POQ}$ and the angle between the tangents $\angle \text{PTQ}$ are supplementary: $\angle \text{POQ} + \angle \text{PTQ} = 180^\circ$ .

Step 1 · Find Angles at the Points of Contact

Question diagram

Since the radius is perpendicular to the tangent at the point of contact: $\text{OP} \perp \text{TP} \implies \angle \text{OPT} = 90^\circ$ $\text{OQ} \perp \text{TQ} \implies \angle \text{OQT} = 90^\circ$

Step 2 · Calculate $\angle \text{PTQ}$ using Angle Sum Property

In quadrilateral $\text{OPTQ}$ , the sum of all interior angles is $360^\circ$ : $\angle \text{POQ} + \angle \text{OQT} + \angle \text{PTQ} + \angle \text{OPT} = 360^\circ$

Substitute the known values: $110^\circ + 90^\circ + \angle \text{PTQ} + 90^\circ = 360^\circ$

$290^\circ + \angle \text{PTQ} = 360^\circ$

$\angle \text{PTQ} = 360^\circ - 290^\circ = 70^\circ$

Answer

(B) $70^\circ$

Common Mistakes

Direct Supplementary Shortcut: Forgetting that the angle subtended by the tangents ( $\angle \text{PTQ}$ ) and the central angle ( $\angle \text{POQ}$ ) are directly supplementary: $\angle \text{PTQ} = 180^\circ - 110^\circ = 70^\circ$ .
Tangent-Radius Angle Error: Assuming the angle between the tangent and radius is something other than $90^\circ$ .

Question 2

More questions in Exercise 10.2