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Circles

NCERT Class 10 · Mathematics Based on NCERT Class 10 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter builds on the basic understanding of circles from Class IX to explore the relationship between a circle and a line, specifically focusing on tangents. The central idea is that a tangent is a special line that touches the circle at exactly one point, and it has two key properties: it is perpendicular to the radius at the point of contact, and the lengths of tangents drawn from an external point are equal. A student must walk away knowing how to identify tangents, apply these two theorems to solve problems, and understand the number of tangents possible from a point inside, on, or outside the circle.
Key Concepts (10)
Circle
A collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre).
e.g. A bicycle wheel, where the centre is the axle and every point on the rim is at the same distance from it.
Non-intersecting line
A line that does not intersect the circle at any point; it has no common point with the circle.
e.g. A line drawn far away from a bangle such that it does not touch or cut the bangle at all.
Secant
A line which intersects the circle at two distinct points.
e.g. A straight line drawn across a ring that cuts through it at two points on the circumference.
Tangent
A line that touches the circle at exactly one point. It is a special case of a secant when the two end points of its corresponding chord coincide.
e.g. The rope on both sides of a pulley, if considered as a ray, is like a tangent to the circle representing the pulley.
Point of contact
The common point of the tangent and the circle.
e.g. The point where a bicycle wheel touches the ground is the point of contact of the tangent (the ground) and the wheel.
Radius
The line segment from the centre of the circle to any point on the circle.
e.g. A spoke of a bicycle wheel is a radius from the centre to the rim.
Chord
A line segment whose endpoints lie on the circle.
e.g. A piece of string stretched across a circular drum from one point on the rim to another.
Length of the tangent
The length of the segment of the tangent from the external point to the point of contact with the circle.
e.g. From a point outside a circular garden, the distance along a straight path that just touches the garden edge to the point of touching.
Normal
The line containing the radius through the point of contact of a tangent.
e.g. At the point where a ladder touches a circular well, the line from the centre to that point is the normal.
Concentric circles
Two or more circles with the same centre but different radii.
e.g. The rings on a target board, all sharing the same centre point.
Formulas & Equations (4)
OP ⊥ XY
Reads as: OP is perpendicular to XY
Variables: O = centre of the circle; P = point of contact; XY = tangent at point P
Use when: Proving that a line is a tangent, or finding distances using right-angle properties in a circle-tangent problem.
⚠ Students often forget to state that the radius must be drawn to the point of contact before applying perpendicularity.
PQ = PR
Reads as: PQ equals PR
Variables: P = external point; Q and R = points of contact of the two tangents from P to the circle
Use when: Finding the length of a tangent from an external point, or proving that two tangent segments are equal.
⚠ Students incorrectly apply this theorem to tangents drawn from a point on the circle (only one tangent exists there).
∠OQP = ∠ORP = 90°
Reads as: Angle OQP and angle ORP are right angles
Variables: O = centre; Q and R = points of contact; P = external point
Use when: Setting up right triangles to apply Pythagoras theorem or RHS congruence in tangent problems.
⚠ Students assume the angle is 90° at the external point P, not at the points of contact Q and R.
OP² = OQ² + PQ²
Reads as: OP squared equals OQ squared plus PQ squared
Variables: O = centre; P = external point; Q = point of contact; OP = distance from centre to external point; OQ = radius; PQ = length of tangent
Use when: Finding the radius or the length of a tangent when the distance from the centre to the external point is known.
⚠ Students mix up which side is the hypotenuse — OP is always the longest side (distance from centre to external point).
How It Works
Existence of a tangent at a point on a circle
  1. Take a circular wire and attach a straight wire AB at a point P so it can rotate about P.
  2. Rotate the straight wire; in most positions it intersects the circle at P and another point Q1, Q2, etc.
  3. In one specific position (A'B'), the wire touches the circle only at point P — this is the tangent.
  4. This shows that at any point on a circle, there is exactly one tangent.
💡 The tangent exists only when the rotating line coincides with the circle at exactly one point; at all other positions it is a secant.
Tangent as a limiting case of a secant
  1. Draw a circle and a secant PQ intersecting it at two points.
  2. Draw lines parallel to the secant on both sides; the chord length gradually decreases.
  3. Continue until the two intersection points come closer and finally coincide.
  4. When the two points coincide, the line becomes a tangent.
💡 A tangent is a secant where the two endpoints of the chord have merged into one — the point of contact.
Number of tangents from a point to a circle
  1. Take a point P inside the circle: any line through P cuts the circle at two points, so no tangent is possible.
  2. Take a point P on the circle: only one tangent can be drawn at that point.
  3. Take a point P outside the circle: exactly two tangents can be drawn to the circle from P.
💡 The number of tangents depends on the position of the point: zero inside, one on, two outside.
Proving tangents from an external point are equal (Theorem 10.2)
  1. Given a circle with centre O, external point P, and tangents PQ and PR at points Q and R.
  2. Join OP, OQ, and OR.
  3. ∠OQP = 90° and ∠ORP = 90° (Theorem 10.1: radius perpendicular to tangent).
  4. In right triangles OQP and ORP: OQ = OR (radii), OP = OP (common).
  5. Therefore, ΔOQP ≅ ΔORP (RHS congruence).
  6. Hence, PQ = PR (CPCT).
💡 The proof uses RHS congruence, not SSS or SAS — students often try the wrong congruence criterion.
Proving the chord of a larger concentric circle is bisected at the point of contact
  1. Given two concentric circles C1 (larger) and C2 (smaller) with centre O.
  2. Chord AB of C1 touches C2 at point P.
  3. Join OP. Since AB is tangent to C2 at P, OP ⊥ AB (Theorem 10.1).
  4. In circle C1, OP is perpendicular to chord AB, so OP bisects AB (perpendicular from centre bisects chord).
  5. Therefore, AP = BP.
💡 This combines two properties: tangent perpendicularity and the chord bisection theorem from Class IX.
Exam Traps — Don't Make These Mistakes
⚠ Confusing secant and tangent
✗ A tangent is a line that touches the circle at one point and a secant is a line that touches the circle at two points.
→ This is partially correct but incomplete — a secant intersects the circle at two points, not 'touches'. The word 'touches' is reserved for tangents.
✓ A secant is a line that intersects the circle at two distinct points. A tangent is a line that touches the circle at exactly one point.
⚠ Misapplying the perpendicular theorem
✗ The tangent is perpendicular to the radius.
→ This is incomplete — the theorem states the tangent is perpendicular to the radius *through the point of contact*. The radius must be drawn to the exact point where the tangent touches.
✓ The tangent at any point of a circle is perpendicular to the radius through the point of contact.
⚠ Assuming two tangents from any external point are equal in all contexts
✗ The lengths of tangents from a point to a circle are equal.
→ This is true only if the point is *outside* the circle. Students often forget to specify 'external point'.
✓ The lengths of tangents drawn from an external point to a circle are equal.
⚠ Incorrectly identifying the point of contact in a diagram
✗ The point where the tangent meets the radius is the point of contact.
→ The point of contact is where the tangent touches the circle, not where it meets the radius. The radius is drawn to that point.
✓ The point of contact is the common point of the tangent and the circle. The radius is drawn from the centre to this point.
⚠ Forgetting the 'only one tangent' at a point on the circle
✗ There can be two tangents at a point on the circle.
→ At a point on the circle, there is exactly one tangent. Two tangents exist only from a point outside the circle.
✓ There is one and only one tangent to a circle passing through a point lying on the circle.
Compare & Contrast
Number of intersection points with the circle
Tangent
Secant
A tangent intersects the circle at exactly one point (point of contact), while a secant intersects the circle at two distinct points.
Number of tangents that can be drawn from the point
Point inside the circle
Point outside the circle
From a point inside the circle, no tangent can be drawn. From a point outside the circle, exactly two tangents can be drawn.
Relationship at the point of contact
Radius
Tangent
The radius is a line segment from the centre to the point of contact; the tangent is a line through the point of contact perpendicular to that radius.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Short Answer question (3 marks), 1 Long Answer question (5 marks) — total approximately 10 marks
Most asked: Theorem 10.1: Tangent is perpendicular to the radius at the point of contact · Theorem 10.2: Lengths of tangents from an external point are equal · Number of tangents from a point (inside, on, outside) · Problems involving concentric circles and tangents · Quadrilateral circumscribing a circle (AB + CD = AD + BC)
Don't skip: Proof-based questions on Theorem 10.1 or Theorem 10.2 (5 marks); Application problems combining tangents with Pythagoras theorem or chord properties (3-5 marks); Quadrilateral circumscribing a circle — proving sum of opposite sides equal (5 marks)
One-Line Revision (read aloud before the exam)
  1. A tangent to a circle is a line that touches the circle at exactly one point.
  2. A secant is a line that intersects the circle at two distinct points.
  3. The tangent at any point of a circle is perpendicular to the radius through the point of contact.
  4. There is exactly one tangent at a point on the circle.
  5. No tangent can be drawn from a point inside the circle.
  6. Exactly two tangents can be drawn from a point outside the circle.
  7. The lengths of tangents drawn from an external point to a circle are equal.
  8. The line containing the radius through the point of contact is called the normal.
  9. A tangent is a special case of a secant when the two endpoints of the chord coincide.
  10. The common point of the tangent and the circle is called the point of contact.
  11. In two concentric circles, the chord of the larger circle touching the smaller circle is bisected at the point of contact.
  12. The centre lies on the angle bisector of the angle between two tangents drawn from an external point.
  13. A quadrilateral circumscribing a circle has the property that the sums of the lengths of opposite sides are equal.
  14. The angle between two tangents drawn from an external point is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

MCQs — 10 Questions with Answers

Q1. A rope wrapped around a pulley makes contact at only one point. Based on Theorem 10.1, why is the rope tangent to the pulley rather than a secant?

  • A. Because a tangent touches a circle at exactly one point, and the radius to that point is perpendicular to the rope ✓
  • B. Because a secant requires two points of contact, but the rope only touches once
  • C. Because the rope is straight, and only straight lines can be tangents
  • D. Because the pulley's centre is not on the rope

Answer: A — A tangent is defined as a line intersecting a circle at exactly one point (Activity 1), and Theorem 10.1 proves the radius at the point of contact is perpendicular to it; option B confuses the definition without connecting to the perpendicularity property that makes this configuration stable.

Q2. A student claims: 'If I draw infinitely many parallel lines, at least three of them will be tangents to the same circle.' Is this claim justified by the chapter content?

  • A. Yes, because a circle is infinitely large
  • B. No, because Activity 2 shows at most two tangents parallel to any given secant ✓
  • C. Yes, because tangents exist at every point on the circle
  • D. No, because parallel lines cannot all be tangents

Answer: B — Activity 2 explicitly demonstrates that when parallel lines are drawn to a secant, the chord lengths decrease until they become zero on both sides, giving exactly two tangents parallel to the secant, not three or more.

Q3. Assertion (A): At any point on a circle, exactly one tangent can be drawn. Reason (R): The tangent at any point of a circle is perpendicular to the radius through the point of contact. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Theorem 10.1 (R) proves the perpendicularity property, which logically enforces that only one tangent can exist at a point—if two distinct tangents existed at P, both would be perpendicular to OP, making them the same line, contradicting the assumption they are distinct.

Q4. Assertion (A): A line inside a circle cannot be a tangent to the circle. Reason (R): Any point on a line inside the circle will either lie inside the circle or on the circle, so the line will always intersect the circle in at least two points. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — A tangent must have exactly one common point with the circle; if a line lies entirely or partially inside, it cannot have exactly one intersection point, so (A) is true; (R) correctly explains why this is impossible using the definition of tangency from the chapter.

Q5. Consider the bicycle wheel in Fig. 10.4. As the wheel rolls forward, the point of contact with the ground constantly changes. Which property ensures that the tangent line (ground) is always perpendicular to the radius at the instantaneous point of contact?

  • A. The wheel's circular shape and Theorem 10.1 ✓
  • B. The fact that the wheel rotates about its centre
  • C. The velocity of the wheel's motion
  • D. The weight of the bicycle pressing the wheel to the ground

Answer: A — Theorem 10.1 establishes that any tangent to a circle (the ground is tangent to the wheel at each contact point) is perpendicular to the radius at that point; the wheel's circular geometry combined with this theorem ensures the property holds regardless of the wheel's motion parameters.

Q6. A student observes that a tangent never crosses the circle. Based on the chapter's definition and Activity 1, why is this observation correct?

  • A. Because a tangent intersects the circle at exactly one point and cannot cross into the interior ✓
  • B. Because the tangent is curved like the circle
  • C. Because tangents are always horizontal or vertical
  • D. Because the centre of the circle repels the tangent

Answer: A — The definition of a tangent (one point of intersection) combined with Theorem 10.1 (perpendicularity to the radius) geometrically prevents the tangent from entering the circle's interior; option B mischaracterizes the tangent as curved when it is always a straight line.

Q7. Two tangents PQ and RS are drawn at points P and S on a circle. If PQ ∥ RS, what can be concluded about the positions of P and S?

  • A. P and S are diametrically opposite points ✓
  • B. P and S could be any two points on the circle
  • C. P and S are separated by an angle less than 90°
  • D. PQ and RS are the same line

Answer: A — From Activity 2 and Theorem 10.1, if two tangents at different points are parallel, they must be perpendicular to parallel radii; parallel radii occur only when they lie on the same line (same diameter), making P and S diametrically opposite.

Q8. Assertion (A): From a point outside a circle, more tangents can be drawn than from a point inside the circle. Reason (R): A point inside the circle lies on no tangent, whereas a point outside the circle can lie on multiple tangents. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — (A) is true based on Activity 3 setup; (R) correctly explains it—from inside no tangent exists, from outside tangents can exist because the point is outside the region swept by all radius-perpendiculars.

Q9. A line touches a circle at exactly one point P. The distance from the circle's centre O to any other point on this line is greater than the distance OP. Which property from the chapter best explains this geometric fact?

  • A. Theorem 10.1: The tangent is perpendicular to the radius at P ✓
  • B. The definition of a circle: all radii are equal
  • C. The property that tangents are straight lines
  • D. Activity 1: the tangent is the limit of secants

Answer: A — The perpendicularity established by Theorem 10.1 makes OP the shortest distance from O to the line (perpendicular is the shortest distance), explaining why all other distances are greater; option B is about the circle itself, not the tangent-line relationship.

Q10. Assertion (A): If PQ is a secant and we draw lines parallel to PQ, eventually both result in tangents, one on each side of PQ. Reason (R): As we move a line parallel to a secant, the chord length decreases and eventually becomes zero, creating a tangent when the two endpoints of the chord coincide. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Activity 2 demonstrates exactly this process: as parallel lines approach the secant PQ from both sides, chord lengths shrink to zero, leaving one tangent on each side; (R) explains the mechanism behind (A) using the tangent-as-limiting-secant concept.

Flashcards

What is a tangent to a circle?

A line that intersects the circle at exactly one point, called the point of contact.

State Theorem 10.1

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

How many tangents can be drawn at a point on a circle?

Exactly one tangent can be drawn at any point on the circle.

What is the difference between a tangent and a secant?

A tangent intersects the circle at one point; a secant intersects the circle at two points.

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

No tangents can be drawn from any point lying inside the circle.

How many tangents can be drawn from a point outside the circle?

Exactly two tangents can be drawn from any point lying outside the circle.

What is the relationship between a tangent and the normal to a circle?

The normal at a point is the line containing the radius through that point; it is perpendicular to the tangent.

In Fig. 10.5, why is Q outside the circle?

If Q were inside the circle, the line XY would intersect the circle at two points and would be a secant, not a tangent.

How many parallel tangents can a circle have at most?

A circle can have at most two parallel tangents on opposite sides of the circle.

What does 'point of contact' mean?

The point of contact is the unique point where a tangent line touches the circle.

Important Board Questions

Define a tangent to a circle and state Theorem 10.1. [2 marks]

A tangent intersects the circle at one point. Theorem 10.1: tangent is perpendicular to the radius at the point of contact.

From a point P outside a circle with centre O and radius 5 cm, two tangents are drawn touching the circle at points A and B. If OP = 13 cm, find the length of tangent PA. [3 marks]

Use Theorem 10.1 to establish OA ⊥ PA. In right triangle OAP, apply Pythagoras theorem: PA² = OP² − OA² = 169 − 25 = 144, so PA = 12 cm.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact (Theorem 10.1). Use the property that OP is the shortest distance from centre O to the tangent line. [5 marks]

Let XY be tangent at P, and Q be any point on XY other than P. Show Q lies outside the circle (if inside, XY would be a secant). Prove OQ > OP for all such Q, meaning OP is the perpendicular distance. Conclude OP ⊥ XY using the shortest distance principle.

Assertion-Reason Questions for Circles

Q1. Assertion (A):

A tangent to a circle can have only one point of intersection with the circle.

Reason (R):

This is because a tangent is defined as a line that touches the circle at exactly one point called the point of contact.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why the assertion is true—the definition of a tangent guarantees exactly one intersection point. This is the foundational property of tangents.

Q2. Assertion (A):

From a point inside a circle, it is not possible to draw any tangent to the circle.

Reason (R):

Because any line passing through an interior point will intersect the circle at two points, making it a secant rather than a tangent.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true and the reason correctly explains the assertion. Activity 3 Case 1 confirms this: interior points cannot have tangents because all lines through them become secants with two intersection points.

Q3. Assertion (A):

The radius of a circle at the point of contact is parallel to the tangent at that point.

Reason (R):

The tangent at any point of a circle is perpendicular to the radius through the point of contact as stated in Theorem 10.1.

Answer: D — A is false but R is true — The assertion is false—the radius is not parallel but perpendicular to the tangent. The reason is true and actually contradicts the assertion. Since the assertion contradicts the reason (and reality), the correct answer is D.

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Case-Based Questions (CBQ) for Circles

Case Study:

A bicycle manufacturer is designing the wheel assembly. The wheel is perfectly circular with a radius of 35 cm and is mounted on an axle at the center. When the bicycle is resting on level ground, the wheel touches the ground at exactly one point. An engineer wants to ensure that the ground provides proper support at the contact point. The ground level is considered horizontal, and the wheel rotates freely about its axle.

Q1. From the perspective of circle geometry, what is the relationship between the wheel (circle) and the ground (line)?

  • A. The ground is a secant to the wheel
  • B. The ground is a tangent to the wheel ✓
  • C. The ground is a chord of the wheel
  • D. The ground is a diameter of the wheel

Answer: B

Q2. At the point where the wheel touches the ground, what is the geometric relationship between the radius of the wheel and the ground surface?

  • A. The radius is parallel to the ground
  • B. The radius is at 45° to the ground
  • C. The radius is perpendicular to the ground ✓
  • D. The radius is at 60° to the ground

Answer: C

Q3. If the axle of the wheel is at a height of 35 cm above the ground, and a vertical line is drawn from the axle perpendicular to the ground, at what distance from this vertical line does the wheel touch the ground?

  • A. 35 cm away
  • B. 0 cm (directly below) ✓
  • C. 70 cm away
  • D. 17.5 cm away

Answer: B

Q4. A rope is held tangent to the wheel at a point on its rim. By which theorem can we prove that this rope is perpendicular to the radius at the point of contact?

  • A. Pythagorean theorem
  • B. Congruence theorem
  • C. Angle sum property of triangles
  • D. Theorem 10.1 (Tangent-Radius theorem) ✓

Answer: D

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Previous Year Questions (PYQ) for Circles

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 31/4/1 · 1 mark

PT is tangent to the circle with centre O and radius 5 cm. OP intersects the circle at Q. If PQ = x, then PT² equals:

  • x² + 5x
  • x² + 10x + 50
  • x² + 10x
  • x² – 25

CBSE 2026 · Set 31/4/1 · 1 mark

In the given figure, PQ and PR are two tangents drawn to a circle with centre O. If ∠ORQ = 25°, then the measure of ∠PQR is:

  • 65°
  • 25°
  • 50°
  • 75°

CBSE 2026 · Set 31/4/1 · 2 marks

In the given figure, AP, AQ and BC are tangents to the circle with centre O. If AB = 6 cm, AC = 7 cm and BC = 5 cm, then what is the length of AP?

CBSE 2026 · Set 31/4/1 · 3 marks

Prove that a parallelogram circumscribing a circle, is a rhombus.

CBSE 2026 · Set 31/4/2 · 3 marks

Two tangents TP and TQ are drawn to a circle with centre O, from an external point T. Prove that ∠PTQ = 2∠OPQ.

CBSE 2026 · Set 31/4/3 · 3 marks

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

CBSE 2026 · Set 31/5/1 · 1 mark

A chord QR subtends an angle of 105° at the centre O of the circle. The measure of ∠RQP is

  • 75°/2
  • 105°/2
  • 75°
  • 15°

CBSE 2026 · Set 31/5/1 · 1 mark

PQ is tangent to a circle at a point P on the circle. The number of tangents which can be drawn to the circle parallel to PQ, is

  • 2
  • 1
  • many
  • zero

CBSE 2026 · Set 31/5/1 · 3 marks

Chord AB of a circle subtends an angle of 120° at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

CBSE 2026 · Set 31/5/1 · 1 mark

In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle.

Find m∠AOP.

CBSE 2026 · Set 31/5/2 · 1 mark

A chord QR subtends an angle of 105º at the centre O of the circle. The measure of ∠RQP is

  • 75º/2
  • 105º/2
  • 75º
  • 15º

CBSE 2026 · Set 31/5/3 · 1 mark

PQ is tangent to the circle with centre O such that OP = 2OQ. m∠OPQ is

  • 15°
  • 60°
  • 45°
  • 30°

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Class 10 Maths Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 10 Mathematics Previous Year Questions, Chapter-wise

All 1107 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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