1. Number of Tangents from a point on a circle
A tangent to a circle intersects the circle at exactly one point. In other words, at any point on the circumference of the circle, there can only be one tangent drawn to the circle. However, if a point outside the circle is considered, it is possible to draw two tangents to the circle, intersecting the circle at diametrically opposite points on the circumference.
Theorem 2
The lengths of tangents drawn from an external point to a circle are equal
i.e., in the given figure, PQ = PR
- In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact
- i.e., AP = PB
- AB is a chord of the outer (bigger) circle and acts as a tangent to the inner (smaller) circle
- No tangent can be drawn to a circle through a point lying inside the circle
- Only one tangent can be drawn to a circle through a point lying on the circle
- Exactly two tangents can be drawn to a circle from a point lying outside the circle.
The word tangent comes from the Latin word 'tangere', which means to touch. A tangent is a line which touches or intersects the circle at only one point. The point where the tangent meets the circle is called the point of contact or the point of tangency.
Scenarios that arise when a circle and a line are given in a plane
- The line PQ does not intersect┬аthe circle.
- Secant to the circle: Two common points A and B that the line PQ and the circle have.
- Tangent to a circle: Only one point A is common to the line PQ and the circle
- There can only be one tangent at a point one the circle. The tangent can be seen as a special case of secant with the common point being called as the point of contact.
- The tangent at any point of a circle is perpendicular to the radius (acting as normal to the circle) through the point of contact.
Theorem1
The tangent at any point of a circle is perpendicular to the radius through the point of contact