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Circles: Tangents, Proofs, and the Missing Figures

This version keeps the same intentional Class 10 maths chapter UI as the other chapters, but restores the key diagrams needed for secants, tangents, equal tangents, and concentric-circle questions.

Introduction to Circles

Definitions, secants, tangents, and what “one point of contact” really means.

A circle is the set of all points in a plane that are at a constant distance from a fixed point called the centre.

In this chapter, the important shift is from general circle vocabulary to tangents, their point of contact, and how a line can meet a circle in 0, 1, or 2 points.

Key Vocabulary

Chord: line segment joining two points on the circle.
Secant: line intersecting the circle at two points.
Tangent: line touching the circle at exactly one point.
Point of contact: the common point between the tangent and the circle.

Line Positions

These are the three visual cases students usually miss: no intersection, secant, and tangent.

Tangent to a Circle

Theorem 10.1: the radius at the point of contact is perpendicular to the tangent.

Formula Focus

OP \perp AB

If $AB$ is tangent at $P$ and $O$ is the centre, then the radius $OP$ is perpendicular to the tangent.

Radius and Tangent

This right angle is the core fact behind almost every tangent-length question.

Solved Example

A tangent $PQ$ touches a circle of radius $5$ cm at $P$ and $OQ = 13$ cm. Find $PQ$ .

Show solution

Since $OP \\perp PQ$ , triangle $OPQ$ is right-angled.

PQ^2 = OQ^2 - OP^2 = 13^2 - 5^2 = 144

So,

PQ = 12 \\text{ cm}

Number of Tangents from a Point

Inside gives 0, on the circle gives 1, outside gives 2 equal tangents.

Three Cases

Point inside

0 tangents

Point on circle

1 tangent

Point outside

2 tangents

This is one of the most useful visual summaries from your reference code.

Theorem 10.2

PA = PB

If tangents from the same external point $P$ touch the circle at $A$ and $B$ , then their lengths are equal.

Equal Tangents

Both tangent segments from one external point have equal length, and each radius is perpendicular to its tangent.

Concentric Circles and Chord Visual

The missing chord-tangent figure used in one of the most common NCERT proof patterns.

Concentric Circles

A chord of the outer circle tangent to the inner circle is bisected at the point of contact.

Concentric Circles Example

Two concentric circles have radii $5$ cm and $3$ cm. Find the length of the chord of the larger circle which is tangent to the smaller circle.

Show solution

The radius to the point of contact is perpendicular to the chord and bisects it.

Half-chord

= \\sqrt{5^2 - 3^2} = 4

So the full chord is

8 \\text{ cm}

Circles Summary

Tangent

Touches the circle at exactly one point.

Radius Rule

The radius at the point of contact is perpendicular to the tangent.

Equal Tangents

Tangents from the same external point are equal.

Angle Relation

The centre angle and the angle between tangents are supplementary.

Inside / On / Outside

0, 1, and 2 tangents respectively.

Concentric Chord

A chord tangent to the inner circle is bisected at contact.

Open Practice

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