How do I know when to use Pythagoras in this chapter?

Use Pythagoras whenever the radius to the point of contact, the tangent length, and the line joining the external point to the centre form a right triangle.

What is the most important side relation in a quadrilateral touching a circle?

If a quadrilateral circumscribes a circle, then the sum of one pair of opposite sides equals the sum of the other pair: AB + CD = BC + AD.

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Detailed Notes • Tangents • Proofs • Visual Geometry

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

Circles: revision sheet

Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.

Recall checklist

Tangents, Secants, and How They Touch a Circle
Theorem: Tangent is Perpendicular to the Radius
Equal Tangents from an External Point
Angle Between Tangent and Chord — and Key Angle Results
Common Mistakes and Board Answer Template

For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.

Formula recap (1)

Equal Tangents from an External Point

AP = AQ

Equal tangents from external point A. Proved by RHS congruence of triangles OAP and OAQ.

Explain your method

Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.

A tangent from a point is 24 cm long and the distance of the point from the centre is 25 cm. Find the radius.

Check worked solution 1

Using the right triangle formed by radius, tangent, and distance to the centre: r^2 = 25^2 - 24^2 = 625 - 576 = 49, so the radius is 7 cm.

Two tangents from an external point are inclined at 80 degrees. Find the angle subtended by the points of contact at the centre.

Check worked solution 2

The angle between tangents and the central angle are supplementary, so the angle at the centre is 180 degrees - 80 degrees = 100 degrees.

Apply the concept, avoid the trap

Common mistake: Using equal tangents for segments drawn from different external points.

Use this reasoning: The equality applies to the two tangent segments drawn from the same external point to a circle. Identify that point before writing the equation.

Worked application

PA and PB are tangents from P to a circle with centre O. If OA = 5 cm and OP = 13 cm, find PA and the perimeter of quadrilateral OAPB.

Reveal the method and checks

OA is perpendicular to tangent PA at A, so triangle OAP is right-angled. PA² = OP² − OA² = 169 − 25 = 144; hence PA = 12 cm.
Tangents from P are equal, so PB = PA = 12 cm. Radii OA and OB are both 5 cm.
The boundary O–A–P–B–O has length 5 + 12 + 12 + 5 = 34 cm. OP is an internal diagonal and must not be included in the perimeter.

Check your understanding

Can a point strictly inside a circle have a real tangent segment to it?

Check your reasoning

No. A line through an interior point intersects the circle twice; it cannot touch the circle at exactly one point. The right-triangle construction above also requires OP greater than the radius.

Turn mistakes into revision

Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.

Practise this chapter