Question

prove that tangent at any point is perpendicular to radius​

Answer 1

Answer:

We know that, among all line segment joining the point O to a point on l, the perpendicular is shortest to l. Now, OB = OC + BC. 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.

Answer 2

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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