Name: Length of Tangent to a Circle from External Point
Uploaded: 2019-07-19
Duration: 3 min 4 s
Description: A tangent is a straight line that touches a circle at any external point. But, how do you calculate the length of a tangent to a circle from an external poin...

Question

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Accepted Answer

Given: A circle $O$ and a tangent $AB$ at a point $P$ .

To prove: $OP ⊥ AB$ .

Construction: Take a point $Q$ , other than $P$ , on the tangent $AB$ . Join $OQ$ .

Proof: Since, $Q$ is a point on the tangent $AB$ , other than the point of contact $P$ , so $Q$ will be outside the circle.

Let $OQ$ intersect the circle at $R$ .

$\Rightarrow OQ = OR + RQ$

$\Rightarrow OQ > OR$

$\Rightarrow OQ > OP [∵ OR = OP = radius]$

$\Rightarrow OP < OQ$

Thus, $OP$ is shorter than any other line segment joining $O$ to any point on $AB$ .

But, among all line segments, joining the point $O$ to a point on $AB$ , the shorter one is the perpendicular from $O$ on $AB$ .

Hence, $OP ⊥ AB$

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Important Questions on Circles

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