1. What is a Tangent?
A tangent to a circle is a straight line that touches the circle at exactly one point. This point of contact is called the point of tangency.
If a line touches the circle only at point \(P\), then the line is tangent at \(P\).
2. Point of Contact
The point of contact is the single point where the tangent meets the circle.
At this point, the tangent just ‘grazes’ the circle instead of cutting through it.
3. Key Property: Tangent is Perpendicular to Radius
The most important property of a tangent is:
The tangent to a circle is perpendicular to the radius at the point of contact.
\(OP \perp \text{tangent at } P\)
This means angle between the tangent and the radius = 90°.
3.1. Why is this useful?
This property helps in solving geometry problems involving right angles, triangles, and distances from the centre.
4. Number of Tangents from a Point
The tangents you can draw depend on where the point lies.
4.1. Point on the Circle
You can draw exactly one tangent from a point lying on the circle.
4.2. Point Outside the Circle
You can draw exactly two tangents from a point outside the circle.
4.3. Point Inside the Circle
No tangent can be drawn from a point inside the circle.
5. Two Tangents from an External Point
If \(P\) is a point outside the circle and two tangents \(PA\) and \(PB\) touch the circle at \(A\) and \(B\), then these tangents have special equalities.
5.1. Equal Tangents Theorem
Tangents drawn from an external point are equal in length.
PA = PB
5.2. Angles Formed
The line joining the centre and the external point bisects the angle between the two tangents.
6. Tangent–Radius Theorem in Problems
Most tangent problems rely on right triangles formed at the point of tangency.
6.1. Example
In a circle with radius \(5\text{ cm}\), a tangent touches the circle at \(T\). If \(O\) is the centre, then:
OT = 5\text{ cm}
Since the tangent is perpendicular to the radius:
\angle OTP = 90^\circ
7. Real-Life Situations with Tangents
Examples where tangents naturally appear:
- A wheel touching the ground at one point
- A ladder leaning just touching a circular pole
- A bicycle tyre touching the road
- Touchpoints in gears and machinery