Important Theorems & Core Concepts
1. Theorem on Perpendicularity of Tangent
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. Theorem on Equality of Tangents
The lengths of tangents drawn from an external point to a circle are equal.
If PA and PB are two tangents drawn from an external point P to a circle:
PA = PB
Points to Remember
Key Property: Radius is perpendicular to the tangent at the point of contact (Radius ⊥ Tangent).
Tangent Definition: A tangent to a circle intersects or touches the circle at exactly one point.
Important Practice Questions
- Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Prove that the lengths of tangents drawn from an external point to a circle are equal.
- Find the lengths of tangents or distances using the Pythagoras theorem.
Quick Revision
Perpendicularity
Radius ⊥ Tangent at the point of contact.
Equal Tangents
PA = PB for tangents drawn from external point P.
Single Point Contact
A tangent touches the circle at only one distinct point.
Pythagoras Relation
OP² = OT² + PT² (where OP is distance from center, OT is radius, PT is tangent length).