Circles: Tangents — Part 1: Tangent and radius, two tangents, touching circles: visual lesson, ICSE Class 10 Mathematics

Lesson

Circles: tangents, Part 1

A tangent touches a circle at exactly one point, and it always meets the radius to that point at a right angle. In this part we prove that, and use it for lengths, angles and touching circles.

In this lesson you will

Prove that a tangent is perpendicular to the radius at the point of contact and use it with Pythagoras
Prove that two tangents from an external point are equal and use the right angles at the points of contact
Find the distance between the centres of circles that touch externally or internally
Find the length of a common tangent of two circles

The tangent $PQ$ touches the circle at $T$ and is perpendicular to the radius $OT$.

Concept

What you will be able to do

Prove that a tangent is perpendicular to the radius
Use it with Pythagoras to find lengths
Prove that two tangents from a point are equal
Work with touching circles and common tangents

Part 1: tangents and touching circles.
Result	In words	Used for
Tangent and radius	the radius to the point of contact is perpendicular to the tangent	right-angled triangles and Pythagoras
Two tangents from a point	equal in length	lengths and isosceles triangles
Touching circles	centres and point of contact are in one line	distance between the centres
Common tangent	right triangle on the line of centres	length of the common tangent

Concept

Tangent and secant

A line that meets a circle at two points is a secant; a line that meets it at one point only is a tangent, and that point is the point of contact.

a chord is the part of a secant inside the circle
a tangent never enters the circle

The line $PQ$ is a tangent (one point $T$); the line through $C$ and $D$ is a secant (two points).

Cookies