Question

Prove that the lengths of two tangent segments drawn to the circle from an external point are equal. 

Answer 1

Given: O is the centre of the circle and P is a point in the exterior of the circle. A and B are the points of contact of the two tangents from P to the circle.

To Prove: PA = PB

Construction: Draw seg OA, seg OB, and seg OP.

Proof: Line AP ⊥ radius OA and line BP ⊥ radius OB   ... (Tangent perpendicular to radius)

∴ `angle"PAO" = angle"PBO" = 90^@`

In right-angled triangles `triangle "OAP"` and `triangle"OBP"`

hypotenuse OP ≅ hypotenuse OP    ...(Common side)

seg OA ≅ seg OB        ...(Radii of the same circle)

`:.triangle"OAP" ≅ triangle"OBP"`    ...(Hypotenuse-side of theorem)

∴ seg PA ≅ seg PB     ...(c.s.c.t.)

∴ PA = PB