Question

In the adjoining figure, seg AB is a diameter of a circle with centre C. Line PQ is a tangent, which touches the circle at point T. Seg AP ⊥ line PQ and seg BQ ⊥ line PQ. Prove that seg CP ≅ seg CQ.

Answer 1

Given: C is the centre of circle. seg AB is the diameter of circle. line PQ is a tangent, seg AP ⊥ line PQ and seg BQ ⊥ line PQ. 

To prove: seg CP ≅ seg CQ 

Construction: Draw seg CT, seg CP and seg CQ. 

Proof: Line PQ is the tangent to the circle at point T. [Given] 

∴ seg CT ⊥ line PQ (i) [Tangent theorem] 

Also, seg AP ⊥ line PQ, seg BQ ⊥ line PQ [Given] 

∴ seg AP || seg CT || 

seg BQ [Lines perpendicular to the same line are parallel to each other] 

∴ AC/CB = PT/TQ [Property of intercepts made by three parallel lines and their transversals] 

But, AC = CB [Radii of the same circle] 

∴ = AC/AC = PT/TQ

∴ PR/TQ = 1 

∴ PT = TQ ………… (ii) 

∴ In ∆CTP and ∆CTQ, 

 seg PT ≅ seg QT [From (ii)] 

∠CTP ≅ ∠CTQ [From (i), each angle is of measure 90° ] 

seg CT ≅ seg CT [Common side] 

∴ ∆CTP ≅ ∆CTQ [SAS test of congruence] 

∴ seg CP ≅ seg CQ [c.s.c.t]