PQ is a chord in the minor segment of a circle and R is a point on the minor arc PQ. The tangents at the points P and Q meet at the point T. If ∠PRQ = 102°, then the measure of ∠PTQ is?

Given:

PQ is a chord in the minor segment of a circle.

R is a point on the minor arc PQ.

Tangents at points P and Q meet at point T.

∠PRQ = 102°

Formula used:

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

The sum of opposite angles in a cyclic quadrilateral is 180°.

The angle between the tangent and the chord through the point of contact is equal to the angle in the alternate segment (Tangent-Chord Theorem).

The sum of angles in a quadrilateral is 360°.

Calculation:

Let O be the center of the circle.

Consider the cyclic quadrilateral formed by P, R, Q and another point S on the major arc PQ.

The angle subtended by the chord PQ at any point in the major segment will be supplementary to ∠PRQ.

The angle subtended by the major arc PQ at the circumference is ∠PRQ = 102°.

The angle subtended by the minor arc PQ at the circumference (on the major segment) would be 180° - 102° = 78°.

∠PSQ = 180° - ∠PRQ (as PRQS is a cyclic quadrilateral if S is on the major arc).

So, ∠PSQ = 180° - 102° = 78°.

The angle subtended by the minor arc PQ at the center O, i.e., ∠POQ, is twice the angle subtended by it at the circumference in the major segment (∠PSQ).

⇒ ∠POQ = 2 × ∠PSQ

⇒ ∠POQ = 2 × 78°

⇒ ∠POQ = 156°

Now, consider the quadrilateral TP OQ. TP and TQ are tangents to the circle at P and Q, respectively.

We know that the radius is perpendicular to the tangent at the point of contact.

The sum of angles in the quadrilateral TP OQ is 360°.

⇒ ∠PTQ + ∠TPO + ∠POQ + ∠TQO = 360°

⇒ ∠PTQ + 90° + 156° + 90° = 360°

⇒ ∠PTQ + 336° = 360°

⇒ ∠PTQ = 360° - 336°

⇒ ∠PTQ = 24°

∴ The correct answer is option 2.