Tangent to circle — prove tangent is perpendicular to radius at point of contact

Question

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

(NCERT Class 10 — Circles, 3-mark board question)

Solution — Step by Step

Let $O$ be the centre of the circle and $P$ be the point of contact. Let $XY$ be the tangent at $P$ . We need to prove $OP \perp XY$ .

Assume $OP$ is NOT perpendicular to $XY$ . Then let $OQ$ be the perpendicular from $O$ to $XY$ , where $Q$ is a point on $XY$ different from $P$ .

Since $OQ \perp XY$ , we know $OQ$ is the shortest distance from $O$ to the line $XY$ . This means:

OQ < OP

But $OP = r$ (radius of the circle). So $OQ < r$ , which means $Q$ lies inside the circle.

Now here is the problem: $Q$ is on the tangent line $XY$ , and it lies inside the circle. This means the tangent line passes through an interior point of the circle, so it must intersect the circle at two points.

But a tangent touches the circle at exactly one point — contradiction!

Our assumption was wrong. Therefore, $OP \perp XY$ .

The tangent at any point of a circle is perpendicular to the radius through the point of contact. $\square$

Why This Works

The proof relies on two facts:

The perpendicular from a point to a line is the shortest distance.
A tangent touches the circle at exactly one point (definition of tangent).

If the radius were not perpendicular, some other segment from the centre to the tangent would be shorter than the radius — placing a point of the tangent inside the circle. That breaks the definition of a tangent.

This theorem has a powerful converse too: if a line is perpendicular to a radius at its endpoint on the circle, then that line is a tangent. This converse is used heavily in construction problems.

Alternative Method

You can also prove this using calculus (for a circle $x^2 + y^2 = r^2$ , the slope of the radius to point $(a, b)$ is $b/a$ , and the tangent slope via implicit differentiation is $-a/b$ ; their product is $-1$ , confirming perpendicularity). But the contradiction method above is what CBSE expects.

This proof is asked almost every year in CBSE 10th boards — either directly or as part of a larger circles question. Memorise the structure: assume not perpendicular → drop perpendicular → shorter than radius → point inside circle → contradiction with tangent definition.

Common Mistake

Students often write “OQ is inside the circle because OQ is less than radius” but forget to explain WHY that creates a contradiction. You must explicitly state: since Q lies inside the circle and is on the tangent, the tangent intersects the circle at two points, contradicting the definition of a tangent. Without this final link, the proof is incomplete and you lose marks.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next