1. Tangents from an External Point
When a point lies outside a circle, you can draw exactly two tangents from that point to the circle. These tangents touch the circle at two distinct points.
If \(P\) is an external point and the tangents touch the circle at \(A\) and \(B\), then \(PA\) and \(PB\) are the two tangents.
2. Equal Tangents Theorem
The most important property is:
The two tangents drawn from an external point to a circle are equal in length.
PA = PB
This theorem helps in constructing isosceles triangles and solving angle-based questions.
2.1. Reason Behind Equality
The radii at the points of tangency are perpendicular to the tangents:
OA \perp PA, \quad OB \perp PB
This creates two congruent right triangles \(\triangle OAP\) and \(\triangle OBP\), giving:
PA = PB
3. Angle Between Tangents
When two tangents are drawn from an external point, they form an angle at that point. This angle has a special relation with the angle subtended by the line joining the points of tangency at the centre.
3.1. Angle Property
The angle between the tangents is supplementary to the angle subtended at the centre by the chord joining the points of tangency.
\angle APB + \angle AOB = 180^\circ
4. Line from Centre Bisects the Angle
The line joining the external point \(P\) and the centre \(O\) bisects the angle formed between the two tangents.
OP \text{ bisects } \angle APB
4.1. Why?
Since \(PA = PB\), triangle \(PAB\) is isosceles. Its symmetry forces the line \(OP\) to bisect the vertex angle.
5. Right Angle Between Radius and Tangent
At each point of tangency, the radius is perpendicular to the tangent:
OA \perp PA, \quad OB \perp PB
This property is used in nearly every tangent-based numerical or construction problem.
6. Example to Understand the Properties
Suppose a point \(P\) is outside the circle and tangents touch the circle at \(A\) and \(B\).
6.1. Example
If the length of one tangent is \(12\text{ cm}\), then the other tangent from the same point is also:
PB = 12\text{ cm}
Also, the angle between the tangents and the centre-line relationships can be applied for further calculations.
7. Real-Life Applications
Tangent properties are used in:
- Engineering gear designs
- Tyre–road contact modelling
- Architectural circle-based constructions
- Optics and reflection paths involving circular mirrors