Prove that there is one and only one tangent at any point on the circumference of a circle. - Mathematics

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Sum

Prove that there is one and only one tangent at any point on the circumference of a circle.

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Solution

Let P be a point on the circumference of a circle with centre O.

If possible, Let PT and PT’ be two tangents at a point P of the circle.

Now, the tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ OP ⊥PT and similarly, OP⊥PT’

⇒ OPT = 90° and ∠OPT’ = 90°

⇒ OPT = ∠OPT’

This is possible only when PT and PT’ coincide. Hence, there is one and only one tangent at any point on the circumference of a circle.

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