i. ∠PQR m(arc PR) [Inscribed angle theorem]
= 1/2 × 140° = 70°
∠PQR is the exterior angle of ∆POQ. [Remote interior angle theorem]
∴ ∠PQR = ∠POQ + ∠QPO [R – Q – O]
∴ 70° = ∠POR + ∠QPO
∴ 70 = 36° + ∠QPO
∴ ∠QPO = 70° – 36° = 340
Now, ray OP is tangent at point P and segment PQ is a secant.
∴ ∠QPO = 1/2 m(arcPQ) [Theorem of angle between tangent and secant]
∴ 34° = 1/2 m(arc PQ)
∴ m(arc PQ) = 68°
ii. Here, OP = 7.2, OQ = 3.2
Line OP is the tangent at point P [Given] and seg OR is the secant.
∴ OP2 = OQ × OR [Tangent secant segments theorem]
∴ 7.22 = 3.2 × OR
∴ 51.84 = 3.2 × OR
∴ OR = 51.84/3.2
∴ OR = 16.2 units
Now, OR = OQ + QR [O – Q – R]
∴ 16.2, = 3.2 + QR
∴ QR = 16.2 – 3.2
∴ QR = 13 units
iii. Here, OP = 7.2, OR = 16.2
OP2 = OQ × OR [Tangent secant segments theorem]
∴ 7.22 = OQ × 16.2
∴ OQ = 51.84/16.2
∴ OQ = 3.2 units
Now, OR = OQ + QR [O – Q – R]
∴ 16.2 = 3.2 + QR
∴ QR = 16.2 – 3.2
∴ QR = 13 units