Question

Prove that "A tangent to a circle is perpendicular to the radius a drawn from the point of contact."

Answer 1

Given : A circle C (0, r) and a tangent l at point A.

To prove : OA ⊥ l

Construction : Take a point B, other than A, on the tangent l. Join OB. Suppose OB meets the circle in C.

Proof: We know that, among all line segment joining the point O to a point on l, the perpendicular is shortest to l.

OA = OC  (Radius of the same circle)

Now, OB = OC + BC.

∴ OB > OC

⇒ OB > OA

⇒ OA < OB

B is an arbitrary point on the tangent l. Thus, OA is shorter than any other line segment joining O to any point on l.

Here, OA ⊥ l

Answer 2

given: o is the centre of the circle and ab is the tangent.

to prove: op perpendicular to ab

proof:

draw OB. when OB lay outside of the circle. when it came inside the circle it become a secant not a tangent.

OB is greater than OP

this will happen in all the point in AB execpt Point p.

therefore, OP perpendicular to OB

hence proved.

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