Theorem 10.1 Class 10 Maths – Circles (NCERT)
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Circle with center O and tangent at point P
Proof (Method 1: By Contradiction)
Let a circle have center O and radius OP.
A tangent is drawn at point P. Suppose it is not perpendicular to OP.
Then, another line through P can be drawn that makes an acute angle with OP. This line will cut the circle again at some point Q, meaning it is a secant, not a tangent. This is a contradiction.
Therefore, the tangent at P must be perpendicular to OP.
Proof (Method 2: Using Right Triangle Property)
Take any line through P that is not perpendicular to OP. Let it intersect the circle at another point Q.
Then, in triangle OQP, we have OQ > OP. So the line meets the circle at two points, which means it is a secant.
Only when the line at P is ⟂ to OP, it touches the circle at exactly one point. Hence proved.
Key Points
- A tangent touches the circle at exactly one point.
- The radius to the point of tangency is always ⟂ to the tangent.
- This theorem is the base for solving all tangent–circle problems.