Learning Objectives
- Understand the concept of a tangent to a circle
- Prove and apply the theorem that a tangent is perpendicular to the radius at the point of contact
- Prove and apply the theorem on lengths of tangents from an external point
- Solve problems involving tangents to circles
Key Concepts
Tangent and Secant
A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact or point of tangency.
A secant is a line that intersects the circle at two points.
Theorem 1: Tangent Perpendicular to Radius
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
If OA is a radius and PA is a tangent at A, then OA ⊥ PA (∠OAP = 90°).
Number of Tangents from a Point
- From a point inside the circle: 0 tangents
- From a point on the circle: 1 tangent
- From a point outside the circle: 2 tangents
Theorem 2: Lengths of Tangents from an External Point
The lengths of tangents drawn from an external point to a circle are equal.
If PA and PB are tangents from external point P to a circle with centre O, then PA = PB. Also, OP bisects ∠APB, and OP bisects ∠AOB.
Summary
A tangent touches a circle at exactly one point and is perpendicular to the radius at that point. From any external point, exactly two tangents can be drawn, and they are equal in length. These properties are fundamental in solving geometric problems involving circles and tangent lines.
Important Terms
- Tangent
- A line that touches a circle at exactly one point
- Point of Contact
- The single point where a tangent touches the circle
- Secant
- A line that intersects a circle at two distinct points
Quick Revision
- Tangent ⊥ Radius at the point of contact
- From an external point, two tangents can be drawn and they are equal in length
- The angle between the two tangents from an external point and the angle subtended by the line segment joining the points of contact at the centre are supplementary
- From a point inside the circle, no tangent can be drawn
- The tangent at the end of a diameter is parallel to the tangent at the other end