In figure 3.96 P is the point of contact. (1) If m(arc PR) = 140°, ∠POR = 36°, find m(arc PQ) (2) If OP = 7.2, OQ = 3.2, find OR and QR (3) If OP = 7.2, OR = 16.2, find QR.

In figure 3.96 P is the point of contact.

(1) If m(arc PR) = \(140^\circ\), \(\angle POR = 36^\circ\), find m(arc PQ)

(2) If OP = 7.2, OQ = 3.2, find OR and QR

(3) If OP = 7.2, OR = 16.2, find QR.

Figure: O is outside the circle; OP is a tangent touching the circle at P and O–Q–R is a secant.

Doubtly Asked 24 Oct 2026 Save WhatsApp Report

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Answer: (1) \(68^\circ\) (2) OR = 16.2, QR = 13 (3) QR = 13

(1) Angle between a tangent and a secant drawn from an external point:

\[\angle POR = \frac{1}{2}[m(\text{arc PR}) - m(\text{arc PQ})]\] \[36 = \frac{1}{2}(140 - m(\text{arc PQ}))\] \[m(\text{arc PQ}) = 140^\circ - 72^\circ = 68^\circ\]

(2) By the tangent secant segments theorem, \(OP^2 = OQ \times OR\):

\[51.84 = 3.2 \times OR,\quad OR = 16.2\] \[QR = 16.2 - 3.2 = 13\]

(3)

\[51.84 = OQ \times 16.2,\quad OQ = 3.2\] \[QR = 16.2 - 3.2 = 13\]

Doubtly AI · 24 Oct 2026 AI-verified · Oct 2026 Report

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In figure 3.96 P is the point of contact. (1) If m(arc PR) = 140°, ∠POR = 36°, find m(arc PQ) (2) If OP = 7.2, OQ = 3.2, find OR and QR (3) If OP = 7.2, OR = 16.2, find QR.

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