Question

Prove that, tangent segments drawn from an external point to the circle are congruent.

Answer 1

Point O is the centre of the circle and point P is external to the circle. Segment PA and segment PB are tangent segments to the circle. Point A and point B are touch points of the tangent segments.

Prove: PA ≅ PB

Construction: Draw OA, OB and OP.

Proof: Each tangent of a circle is perpendicular to the radius drawn through the point of contact  ......(Theorem)

∴ Radius OA ⊥ AP and, Radius OB ⊥ BP  .....(i)

∴ m∠PAO = 90° and m∠PBO = 90°

∴ ΔPAO and ΔPBO are right-angled triangles.

Now in ΔPAO and ΔPBO,

OA = OB  ......(∵ Radius of same circle)

∴ ∠PAO = ∠PBO  ......[Using (i)]

Hypotenuse OP = Hypotenuse OP  ......(∵ Common side)

∴ ΔPAO ≅ ΔPBO  .....(RHS congruency criterion)

∴ line PA ≅ line PB   .....(∵ Corresponding sides of congruent triangles)

Line PA and line PB are tangent.

Hence proved.