Question

Prove that a tangent to a circle is perpendicular to the radius to the point of contact. i.e in the figure given below, if AB is a tangent to the circle C(O,r) at P which is the point of contact, then OP perpendicular AB.

Answer 1

GIVEN:

• A circle C(O,r ) and a tangent XY at a point P.

TO PROVE:

• OP ⊥ XY

CONSTRUCTION:

• Take a point x, other than P, on the Tangent XY. Join OQ. Suppose OQ meets the circle at R.

PROOF:

We know that among all line segments joining the points O to a point on XY, the shortest one is Perpendicular to XY . So, to prove that OP ⊥ XY , it is sufficient to prove that OP is shorter than any other segment joining O to any point of XY .

Clearly,

⠀⠀⠀⠀$\sf{ OP = OR }$ [Radii of the circle]

Now,

⠀⠀⠀⠀$\sf{ OQ = OR + RQ}$

$\sf{\implies OQ > OR}$

$\sf{\implies OQ > OP}$ [ ∵ OP = OR]

$\sf{\implies OP < OQ}$

Thus, OP is shorter than any other segment joining O to any point of XY.

Hence,

⠀⠀⠀⠀OP ⊥ XY

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Proved...⠀⠀⠀⠀⠀