Question

In the adjoining figure, P is the point of contact. 

i. If m (arc PR) = 140° , L POR = 36° , find m (arc PQ) 

ii. If OP = 7.2, OQ = 3.2, find OR and QR 

iii. If OP = 7.2, OR = 16.2, find QR.

Answer 1

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