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Class 10 · Chapter 10 · Mathematics

Circles Cheat Sheet

Theorems

Tangent is perpendicular to the radius

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

OT⊥PTOT \perp PT

In a circle with centre O, PT is the tangent at P; the radius OP meets the tangent at right angles.

Lengths of tangents from an external point

The lengths of the two tangents drawn from an external point to a circle are equal.

PA=PBPA = PB

P is an external point and PA, PB are tangents touching the circle at A and B.

Converse of the tangent–radius theorem

A line drawn through the end of a radius and perpendicular to it is a tangent to the circle.

This is used to construct a tangent and to prove that a given line touches a circle.

Definitions

Circle
The set of all points in a plane at a fixed distance from a fixed point called the centre.
Radius
The distance from the centre of a circle to any point on it.
Chord
A line segment joining two points on a circle.
Diameter
A chord that passes through the centre; the longest chord, equal to twice the radius.
Secant
A line that intersects the circle in two distinct points.
Tangent
A line that touches the circle at exactly one point.
Point of contact
The point at which a tangent touches the circle.
Length of a tangent
The distance from an external point to the point of contact.

Key Points

  • A tangent touches the circle at exactly one point, whereas a secant cuts it at two points.
  • The tangent at any point of a circle is perpendicular to the radius at the point of contact.
  • The lengths of the two tangents drawn from an external point to a circle are equal.
  • At any point on a circle there is exactly one tangent.
  • From a point inside the circle no tangent can be drawn; from a point on the circle exactly one; from an external point exactly two.
  • A tangent can be constructed at a point on the circle by drawing a perpendicular to the radius there.
  • If two tangents PA and PB are drawn from an external point P, then OP bisects the angle APB and also the angle AOB.