CBSE Class 10 Maths Chapter 10 Circles: NCERT Solutions, Theorems, Important Questions & Notes PDF

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CBSE Class 10 Maths Notes – Chapter 10 Circles

Published on February 15, 2026 | Updated on February 15, 2026 | Class 10 Maths

Welcome to these comprehensive CBSE Class 10 Notes for Mathematics Chapter 10: Circles. If you are preparing for your CBSE Board exams or NCERT-based school exams, these notes provide everything you need—from basic definitions to advanced proof-based questions. CBSE Class 10 Maths Notes like these are essential tools for scoring 90+ in your mathematics paper.

1️⃣ Introduction to Circles

What is a Circle?

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

Key Terms You Must Know

Term	Definition
Radius	Distance from centre to any point on the circle
Diameter	Longest chord passing through the centre (2 × Radius)
Chord	Line segment joining two points on the circle
Tangent	A line that touches the circle at exactly one point
Secant	A line that intersects the circle at two distinct points

Importance in CBSE Board Exam

• Weightage: Approximately 3–5 marks
• Question Types: 1-mark MCQs, 2-mark short answers, 4-mark proof-based questions
• NCERT Alignment: 100% based on NCERT Class 10 Maths Book (Chapter 10)

These CBSE Class 10 Notes will help you master every concept required for your board examination.

2️⃣ Chapter Overview

Here is what you will learn in CBSE Class 10 Maths Notes Chapter 10:

• ✅ Tangent to a circle – Definition and properties
• ✅ Number of tangents from a point to a circle
• ✅ Two important theorems related to tangents (with proofs)
• ✅ Proof-based questions (Very Important for 4-5 marks)
• ✅ Numerical problems involving tangent lengths and angles

3️⃣ Key Concepts (NCERT Accurate)

Tangent to a Circle

A tangent to a circle is a line that intersects the circle at exactly one point. This point is called the point of contact.

Figure 1: Tangent perpendicular to radius at point of contact

Theorem 1: Tangent ⟂ Radius Theorem

Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Given: A circle with centre O, radius OP, and line XY touching the circle at point P.
To Prove: OP ⟂ XY

Proof:

1. Take any point Q on XY other than P.
2. Q lies outside the circle (since XY touches circle only at P).
3. Therefore, OQ > OP (radius is shortest distance from centre to tangent line).
4. This is true for all points on XY except P.
5. Hence, OP is the shortest distance from O to XY.
6. Therefore, OP ⟂ XY. [Proved]

Figure 2: Proof of Tangent perpendicular to Radius Theorem

Theorem 2: Equal Tangents Theorem

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

Given: A circle with centre O, external point P, and two tangents PQ and PR touching the circle at Q and R respectively.
To Prove: PQ = PR

Proof:

1. Join OQ, OR, and OP.
2. In ΔOQP and ΔORP:
   • OQ = OR (Radii of same circle)
   • OP = OP (Common side)
   • ∠OQP = ∠ORP = 90° (Radius ⟂ Tangent)
3. Therefore, ΔOQP ≅ ΔORP (by RHS congruence rule)
4. Hence, PQ = PR (CPCT). [Proved]

Figure 3: Equal Tangents from External Point Theorem

Number of Tangents from a Point

Position of Point	Number of Tangents	Explanation
Inside the circle	0	No tangent possible as line will intersect at two points
On the circle	1	Exactly one tangent perpendicular to radius
Outside the circle	2	Two equal length tangents possible

Figure 4: Construction of Tangents from External Point

4️⃣ Important Theorems (Quick Reference Box)

Theorem 1: Tangent Perpendicular to Radius

• When to use: When you need to prove a right angle or find angle measures
• Key words: "Point of contact," "perpendicular," "90°"
• Marks: Usually 2-3 marks in board exams

Theorem 2: Equal Length Tangents

• When to use: When two tangents are drawn from same external point
• Key words: "External point," "equal lengths," "PQ = PR"
• Marks: Usually 4-5 marks proof questions

5️⃣ Solved Examples (CBSE Pattern)

Example 1: Easy (1 Mark)

Question: How many tangents can be drawn from a point lying inside the circle?

Solution: Zero (0) tangents can be drawn from a point inside the circle.

Reason: Any line through an interior point will intersect the circle at two points, making it a secant, not a tangent.

Example 2: Moderate (2 Marks)

Question: In the figure, if TP and TQ are two tangents to a circle with centre O, and ∠POQ = 110°, find ∠PTQ.

Solution:

• Since TP and TQ are tangents, ∠OPT = ∠OQT = 90°
• In quadrilateral OPTQ: ∠OPT + ∠OQT + ∠POQ + ∠PTQ = 360°
• 90° + 90° + 110° + ∠PTQ = 360°
• 290° + ∠PTQ = 360°
• ∠PTQ = 70°

Example 3: Board-Level Proof (4 Marks) ★

Question: Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Given: AB is diameter of circle with centre O. PQ is tangent at A and RS is tangent at B.
To Prove: PQ || RS

Proof:

1. OA ⟂ PQ (Radius ⟂ Tangent) ... (i)
2. OB ⟂ RS (Radius ⟂ Tangent) ... (ii)
3. AB is straight line, so ∠OAP = ∠OBS = 90°
4. These are alternate interior angles
5. Therefore, PQ || RS [Proved]

Example 4: Numerical Problem (3 Marks)

Question: From a point P, 13 cm from the centre O of a circle of radius 5 cm, two tangents PQ and PR are drawn. Find the length of each tangent.

Solution:

• OP = 13 cm, OQ = 5 cm (radius)
• In right ΔOQP: OP² = OQ² + PQ²
• 13² = 5² + PQ²
• 169 = 25 + PQ²
• PQ² = 144
• PQ = 12 cm
• Therefore, length of each tangent = 12 cm

6️⃣ Smart Tricks for Board Exams

Diagram Labelling Trick

• Always label the centre as 'O'
• Mark point of contact clearly
• Show right angle symbol (90°) where radius meets tangent
• Label external points as P, Q, R

Right Triangle Identification Shortcut

Whenever you see tangent + radius, immediately think RIGHT TRIANGLE

Use Pythagoras theorem: (Tangent)² + (Radius)² = (Distance from external point to centre)²

Tangent Length Solving Trick

• If two tangents from external point P touch at A and B, then PA = PB
• Join external point to centre – this line bisects the angle between tangents

Proof Presentation Trick

• Always write "Given," "To Prove," and "Proof" clearly
• Give reasons in brackets [Radius ⟂ Tangent], [CPCT], etc.
• Use proper geometric symbols (≅, ⟂, ∴)

Time-Saving Strategy

• For 1-mark questions: Just write the answer with brief reasoning
• For 4-mark proofs: Draw neat diagram (1 min), write steps clearly
• Remember: Neatness = Better marks in CBSE

7️⃣ Visual Learning: Key Diagrams

Diagram 1: Basic Circle Elements

Figure 5: Parts of a Circle - Radius, Diameter, Chord, Tangent

Diagram 2: Equal Tangents Proof

Figure 6: Detailed diagram for Equal Tangents Theorem Proof

8️⃣ Most Important Board Questions

⭐ 1 Mark Questions

1. Define tangent to a circle.
2. How many tangents can be drawn from a point on the circle?
3. What is the angle between radius and tangent at point of contact?

⭐⭐ 2–3 Mark Questions

1. Prove that tangent at any point is perpendicular to radius.
2. Find length of tangent from point 10 cm away from centre (radius = 6 cm).
3. In figure, if ∠PTQ = 60°, find ∠POQ.

⭐⭐⭐ 4–5 Mark Proof Questions

1. Prove that lengths of tangents from external point are equal.
2. Prove that tangents at ends of diameter are parallel.
3. Two tangents PQ and PR are drawn from external point P. Prove that ∠QPR + ∠QOR = 180°.

⭐⭐⭐ Case-Study Based Question (New Pattern)

Scenario: A Ferris wheel (circle) has radius 20 m. Two support cables from a point on ground touch the wheel at two points.

Questions:

• Find length of each cable if point is 25 m from centre
• Find angle between the two cables

9️⃣ Common Mistakes to Avoid

Mistake	Correction
Not writing theorem statement	Always write complete theorem statement
Missing right angle symbol	Mark 90° clearly where radius meets tangent
Incomplete proof steps	Write "Given," "To Prove," "Proof" sections
Calculation errors in Pythagoras	Double-check: Tangent² = Hypotenuse² – Radius²
Forgetting CPCT in proofs	Always mention CPCT for congruent triangles

🔟 Practice Section

MCQs (Choose the correct answer)

Q1. The maximum number of tangents that can be drawn from an external point to a circle is:

(a) 0 (b) 1 (c) 2 (d) Infinite

Answer: (c) 2

Q2. A tangent to a circle intersects it in:

(a) 0 points (b) 1 point (c) 2 points (d) 3 points

Answer: (b) 1 point

Assertion-Reason Questions

Q3.

Assertion (A): The tangent at any point of a circle is perpendicular to the radius.

Reason (R): The lengths of tangents from an external point are equal.

(a) Both A and R are true, R explains A (b) Both A and R are true, R does not explain A (c) A is true, R is false (d) A is false, R is true

Answer: (b)

HOTS Question

Q4. Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Solution: The chord is tangent to smaller circle. Distance from centre = 3 cm.

• Half chord = √(5² – 3²) = √16 = 4 cm
• Full chord = 8 cm

🔹 Frequently Asked Questions (FAQ)

Is Circles important for CBSE Class 10 board exam?

Yes, absolutely. Circles (Chapter 10) carries 3-5 marks weightage. The proof-based questions from this chapter are frequently asked in board exams. These CBSE Class 10 Notes cover all important concepts.

How to write proof questions correctly in board exam?

Follow this structure:

1. Draw neat labelled diagram
2. Write "Given" and "To Prove" clearly
3. Write "Construction" (if needed)
4. Write "Proof" with proper reasoning in brackets
5. End with "Hence Proved"

Are these CBSE Class 10 Maths Notes based on NCERT?

Yes, 100%. These notes strictly follow NCERT Class 10 Maths Book Chapter 10. All theorems, proofs, and examples are NCERT-aligned and perfect for CBSE board preparation.

What is the best way to revise Circles before exam?

• Revise both theorems with proofs
• Practice 4-mark questions from previous year papers
• Memorize the trick: "Radius ⟂ Tangent" and "Equal Tangents"
• Use these CBSE Class 10 Notes for quick revision

🔹 Conclusion

Mastering Chapter 10 Circles is essential for scoring well in your CBSE Class 10 Mathematics board exam. Focus on understanding the two main theorems, practice proof-based questions, and avoid common calculation mistakes. These CBSE Class 10 Maths Notes provide you with everything needed for comprehensive preparation.

Next Recommended Read: Constructions – CBSE Class 10 Notes Chapter 11

All the best for your CBSE Board Exams! 🎯

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Last Updated: February 15, 2026 | Reading Time: 15 minutes