In the adjoining figure, seg AB is a diameter of a circle with centre

1.	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» 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]

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»

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]