Chapter 10 of 14

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Kerala (SCERT)Class 10 Mathematics★ Unit IV — Geometry: 15 marks, shared with Triangles · CBSE Class 10 Maths (Chapter 10)

Circles

Circles — tangent to a circle, properties of tangents, number of tangents from external point, lengths of tangents. CBSE Class 10 Maths Chapter 10.

⏱ 20 min readEasy difficulty✓ Free · NCERT-aligned

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Key points, glossary and the questions most likely to show up.

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By the end of this chapter you'll be able to…

1Tell apart a tangent, a secant, and a non-intersecting line to a circle
2Use the fact that a tangent is perpendicular to the radius at the point of contact
3Use the equal-tangents theorem to solve for unknown lengths and angles
4Recognise quadrilaterals and triangles circumscribing a circle, and apply equal tangent lengths from each vertex
5Justify an MCQ answer geometrically, not just state it

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Why this chapter matters

This chapter has exactly two theorems, and between them they answer almost everything a board paper asks about circles: why a tangent and a radius meet at a right angle, and why two tangents from the same outside point are always equal. Nearly every question in Exercise 10.2 is one of those two facts wearing a different figure.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Applications of Trigonometry

"A circle is the simplest yet most profound shape — perfect symmetry around a single point."

1. About the Chapter

After studying circles in Class 9 (basic definitions, chord properties), Class 10 focuses on TANGENTS to circles — lines that touch a circle at exactly one point.

Why Important

Tangent properties are used in:

Wheel and gear systems
Optics (light rays at tangent angles)
Engineering designs
Astronomy (planetary orbits)

2. Recap — Circle Basics

Definitions

Circle: set of all points equidistant from a fixed point (centre)
Radius: distance from centre to circle
Chord: line segment with both endpoints on the circle
Diameter: longest chord, passes through centre = 2 × radius
Arc: portion of the circumference
Sector: pie-slice region

Position of a Line Relative to a Circle

A line can be:

Non-intersecting (does not touch circle)
Tangent (touches at exactly 1 point)
Secant (intersects at 2 points)

3. Tangent to a Circle

Definition

A tangent is a line that touches a circle at EXACTLY ONE POINT.

Point of Contact

The single point where tangent touches the circle.

Common Examples

Tyre touching the road
Coin balanced on edge of table
Sun's rays just grazing the horizon

4. Properties of Tangents (KEY THEOREMS)

Theorem 1: Tangent is Perpendicular to Radius at Point of Contact

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

If OP is radius to point P on circle, and AB is tangent at P, then OP ⊥ AB.

Proof (Outline)

Suppose tangent AB touches circle at P. Among all line segments from O to AB, OP is the shortest (since others go from O to points outside circle). Shortest distance from a point to a line is perpendicular. So OP ⊥ AB.

Theorem 2: Tangents from External Point are Equal

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

If P is external, and PA, PB are tangents (touching circle at A, B), then PA = PB.

Proof (Outline)

OA = OB (both radii)
∠OAP = ∠OBP = 90° (tangent ⊥ radius)
OP common to both △OAP and △OBP
△OAP ≅ △OBP (RHS criterion)
Therefore PA = PB

Number of Tangents from a Point

Inside circle: 0 tangents
On circle: 1 tangent
Outside circle: 2 tangents

5. Worked Examples

Example 1: Find Tangent Length

A point P is at a distance of 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P to the circle.

Setup:

OP = 13 cm (distance from centre to external point)
OQ = 5 cm (radius to tangent point Q)
OQ ⊥ PQ (tangent perpendicular to radius)

Apply Pythagoras in △OPQ:

PQ² = OP² − OQ² = 169 − 25 = 144
PQ = 12 cm

Recognise the 5-12-13 triple.

Example 2: Equal Tangents

From external point P, two tangents PA and PB are drawn to a circle. If PA = 7 cm, find PB.

By theorem: PA = PB
PB = 7 cm

Example 3: Inscribed Quadrilateral

A circle is inscribed in a triangle ABC, touching sides at D, E, F. If AB = 10, BC = 11, AC = 13, find lengths of tangents.

Let tangents from each vertex = x, y, z.

AD = AF = x
BD = BE = y
CE = CF = z

Then:

AB = x + y = 10
BC = y + z = 11
AC = x + z = 13

Adding: 2(x+y+z) = 34 → x+y+z = 17

z = 17 − 10 = 7
x = 17 − 11 = 6
y = 17 − 13 = 4

Example 4: Tangent Equation Style

Two tangents from external point to circle are 8 cm. Find the radius if the point is 10 cm from centre.

Tangent length = 8
Distance from centre = 10
Radius² = 10² − 8² = 100 − 64 = 36
Radius = 6 cm

6. Real-World Applications

Engineering

Pulley systems use tangent properties
Gear designs based on circles and tangents
Belt drives between two pulleys

Optics

Light rays reflecting off lenses use tangent geometry
Spectacle lenses designed with tangents

Astronomy

Planetary orbit tangents
Eclipses involve tangent lines from Sun to Earth/Moon

Construction

Road curves designed using tangent geometry
Arches in architecture

Modern Tech

GPS uses tangent properties for distance
Robotics uses circular motion with tangents

7. Common Mistakes

Forgetting perpendicularity
- At point of contact, TANGENT IS PERPENDICULAR to radius.
Tangent passes through centre
- NO. Tangent is OUTSIDE circle (except at point of contact).
Tangents from interior point
- From point INSIDE circle: NO tangent possible.
Length confusion
- Tangent length is measured FROM external point TO point of contact.
Pythagoras error
- In △OPQ: OP is hypotenuse (distance to external point), OQ is leg (radius), PQ is leg (tangent length).

8. Indian Context

Circles in Indian Mathematics

Aryabhata approximated π to 3.1416
Madhava of Kerala (14th c.) gave infinite series for π
Indian Vedic geometry used circles extensively

Modern Use

Indian Railways tracks use circular curves
Highway design uses tangent transitions

9. Worked Example with Two Tangents

Example: Tangents from External Point

From external point P, two tangents are drawn touching circle at A and B. If angle APB = 60°, find angle AOB.

In quadrilateral OAPB:

∠OAP = 90° (tangent ⊥ radius)
∠OBP = 90° (tangent ⊥ radius)
∠APB = 60° (given)
Sum of angles in quadrilateral = 360°
∠AOB = 360° − 90° − 90° − 60° = 120°

10. Conclusion

The geometry of tangents is elegant and PROFOUND:

Tangent ⊥ Radius at point of contact
External tangents are EQUAL in length
Used everywhere — engineering, optics, design

Master:

Two key theorems
Pythagoras in tangent problems
Equal tangent lengths

This chapter is high-yield for board exams. Practice 15+ problems.

Circles and tangents: geometry's most graceful relationship.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Tangent ⊥ radius

OP ⊥ tangent at P, where O is the centre

Theorem 10.1 — the single most-used fact in this chapter

Equal tangent lengths

PA = PB, for tangents from an external point P touching at A and B

Theorem 10.2 — proved by congruent right triangles or by Pythagoras

Tangent length formula

PA = √(OP² − r²)

From the right triangle OAP, since ∠OAP = 90°

Angle bisector property

OP bisects ∠APB

Because △OAP ≅ △OBP — the centre lies on the bisector of the angle between the two tangents

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Forgetting to justify an MCQ answer

✓ Exercise 10.2 Q1-Q3 explicitly say 'choose the correct option and give justification' — the reasoning carries the marks, the letter alone does not.

WATCH OUT

✗ Treating any line touching a circle as tangent at that point

✓ A tangent meets the circle in exactly one point. If a line crosses it at two points it is a secant, however close together those points look in a rough sketch.

WATCH OUT

✗ Not using the perpendicularity when it is not explicitly given

✓ Any time a tangent and a radius to the same point of contact appear together, ∠(radius, tangent) = 90° is available for free — many proofs stall because this was not marked on the figure.

WATCH OUT

✗ Assuming a circumscribing quadrilateral is uniquely determined by two sides

✓ In circumscribing-quadrilateral proofs, the equal tangent lengths from each vertex are the four unknowns to name; the theorem being proved (like AB + CD = AD + BC) says how they combine, not what they individually equal.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 10.1

Exercise 10.1

Tangent, secant, non-intersecting line — definitions and a first tangent-length calculation

Questions

Ex 10.2

Exercise 10.2

Applying the two theorems — tangent lengths, MCQs with justification, and circumscribing shapes

Questions

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Circles?

3 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

3 questions~2 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Three positions of a line relative to a circle: non-intersecting, secant (2 points), tangent (1 point)
•A tangent is a limiting case of a secant, as the two intersection points merge into one
•Theorem 10.1: the tangent at a point is perpendicular to the radius through that point
•One and only one tangent exists at any point of a circle
•From a point inside a circle: no tangent. On the circle: exactly one. Outside: exactly two.
•Theorem 10.2: the two tangent lengths from an external point are equal
•The centre lies on the bisector of the angle between two tangents from the same external point
•This chapter has two exercises, 10.1 (4 Q) and 10.2 (13 Q) — 17 questions in all

Kerala (SCERT) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8

Question type	Marks each	Typical count	What it tests
MCQ	1	2	Properties
Short	2-3	1-2	Tangent length
Long	5	0-1	Inscribed circle problems

Prep strategy

Memorise 2 key theorems
Master Pythagoras in tangent problems
Practice inscribed circle problems

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Wheels and pulleys

Tyre on road, belt on pulley use tangent geometry.

Optics

Light rays reflecting from circles/spheres follow tangent geometry.

Road design

Curved roads transition via tangent lines.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Mark every radius-to-point-of-contact segment on your figure first — that is where the right angle lives

In circumscribing-shape proofs, name the four tangent lengths from the four vertices before writing anything else

For MCQs, write the one or two lines of reasoning even though only the letter is asked

When two tangents from one point appear, mark them equal immediately — it is usually the key step

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

STRETCH

Power of a point

STRETCH

Pole and polar

STRETCH

Conic sections

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 10 BoardHigh

Maths OlympiadHigh

JEE FoundationHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

A chord is the line *segment* joining two points on a circle. A secant is the *full line* through those same two points, extended in both directions. A tangent touches the circle at exactly one point and does not cross into the interior at all. So a chord becomes a secant when extended, and a tangent is what a secant turns into when its two points of intersection merge into one — the textbook's own Activity 2 shows this by sliding a secant until it just grazes the circle.

Both are explicitly examinable and have appeared as standalone proof questions in board papers, so know the proofs, not just the statements. Theorem 10.1 is proved by showing OP is the shortest distance from the centre to the tangent line; Theorem 10.2 is proved with congruent right triangles (RHS) or, equivalently, with Pythagoras.

No. Circles is one of the few Class 10 chapters the rationalisation left untouched — still two exercises, 4 and 13 questions, and the same two theorems in the summary. If a source tells you otherwise for this chapter, it is wrong.

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