ICSE Class 10 Maths Chapter 18 Tangents and Intersecting Chords

Chapter 18: Tangents And Intersecting Chords

18.1 Introduction

If a circle and a straight line are drawn in a plane, then with respect to each other, they may have one of the following three positions:

First position: The line does not meet (cut) the circle.

Second position: The line cuts the circle at two points.

Third position: The line meets (touches) the circle at one point only.

The straight line which cuts a circle at two points is called the secant of the circle. In the given figure, CD is a secant of the given circle.

The line which touches a circle at one point only is called the tangent of the circle. In the given figure, EF is tangent to the given circle at point P of the circle.

The point at which a tangent touches the circle is called the point of contact.

Theorem 10

The tangent at any point of a circle and the radius through this point are perpendicular to each other.

Given: A circle with centre O. AB is a tangent to the circle at point P and OP is the radius of the circle.

To Prove: OP - AB.

Construction: Take a point Q (other than P) on the tangent AB. Join OQ.

Remember: Out of all the line segments drawn from a given point to a given line, the perpendicular is the shortest.

Proof

Hence Proved.

Remember:

1. No tangent can be drawn to a circle through a point inside the circle.

2. One and only one tangent can be drawn through a point on the circumference of the circle.

3. Only two tangents can be drawn to a circle through a point outside the circle.

Corollary

If two tangents are drawn to a circle from an exterior point (the point which lies outside the circle):

(i) the tangents are equal in length;

(ii) the tangents subtend equal angles at the centre of the circle and

(iii) the tangents are equally inclined to the line joining the point and the centre of the circle.

Given: A circle with centre O. PA and PB are two tangents drawn to this circle, from an exterior point P.

To Prove:

(i) PA = PB,

(ii) - AOP = - BOP,

(iii) - APO = - BPO.

Proof:

Hence Proved.

Theorem 11

If two circles touch each other, the point of contact lies on the straight line through the centres.

Case I: When the given two circles touch each other externally.

Given: Two circles with centres A and B touching each other externally at point P.

To Prove: P lies on the line AB.

Construction: Through the point of contact P, draw a common tangent PQ. Join AP and BP.

Proof:

Hence Proved.

Case II: When the given two circles touch each other internally.

Given: Two circles with centres A and B touching each other internally at point P.

To Prove: P lies on the line AB produced.

Construction: Through the point of contact P, draw a common tangent PQ. Join AP and BP.

Proof:

Hence Proved.

Remember:

1. If r1 and r2 be radii of two circles touching each other at a point and d be the distance between their centres then:

(i) d = r1 + r2 when circles touch each other externally,

(ii) d = r1 - r2 when circles touch each other internally. i.e. d = r1 - r2 when r1 is greater and d = r2 - r1 when r2 is greater.

2. If AB and CD are tangents to the same circle at points P and Q such that AB is parallel to CD, then PQ is always the diameter of that circle.

3. Concentric circles means circles with the same centre.

Teacher's Note

Tangent and radius concepts appear in everyday applications like designing wheel rims and understanding how a ladder leans against a circular tank.

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