Circles Class 10
Download the PDF containing Circles Class 10 notes and a mind map. Revise tangents, points of contact and the two main tangent theorems with clear explanations, proofs, solved examples and student FAQs.
Why does a tangent form a right angle with a radius? Why are the two tangent lengths from an external point equal? Class 10 Circles develops these ideas and shows how to use them in length calculations and geometry proofs. Recognising the point of contact is often the first step towards solving a question.
These Circles Class 10 notes support your 2026 revision with theorem statements, important relationships and step-by-step examples. Read the explanations first, use the Circles Class 10 mind map for quick recall, and then practise questions independently. State the reason behind each equality rather than relying on the appearance of a diagram.
Circles Class 10: Important Terms
| Term | Meaning | Remember |
|---|---|---|
| Radius | A segment joining the centre to a point on the circle. | All radii of the same circle are equal. |
| Chord | A segment joining two points on the circle. | A diameter is a chord through the centre. |
| Secant | A line intersecting the circle at two distinct points. | It passes through the circle. |
| Tangent | A line meeting the circle at exactly one point. | It is perpendicular to the radius at that point. |
| Point of contact | The point where the tangent meets the circle. | Join this point to the centre. |
| Tangent length | The length from an external point to the point of contact. | The tangent line itself extends indefinitely. |
How Many Tangents Can Be Drawn from a Point?
- Inside the circle: No tangent can be drawn.
- On the circle: Exactly one tangent can be drawn.
- Outside the circle: Exactly two tangents can be drawn.
If the circle has centre O and radius r, compare OP with r. OP < r means P is inside, OP = r means P is on the circle, and OP > r means P is outside.
Circles Class 10 Theorem 10.1
The Tangent Is Perpendicular to the Radius
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
If a line is tangent at A to a circle with centre O, then OA is perpendicular to that tangent. An external point P on the tangent therefore gives ∠OAP = 90°.
Proof Using the Shortest Distance
Let the tangent touch the circle at A, and choose any other point Q on the tangent. Since the tangent meets the circle only at A, Q lies outside the circle. Therefore, OQ > OA.
OA is thus the shortest distance from O to the tangent line. The shortest distance from a point to a line is perpendicular. Hence OA is perpendicular to the tangent at A.
Join the Centre to the Contact Point
When a tangent is given, drawing the radius to its point of contact creates a right angle. This often reveals a right triangle suitable for Pythagoras’ theorem.
Circles Class 10 Theorem 10.2
Tangents from an External Point Have Equal Lengths
The lengths of tangents drawn from an external point to a circle are equal.
If PA and PB are tangent segments from the same external point P, then PA = PB.
Proof Using RHS Congruence
Join OA, OB and OP, where O is the centre and A and B are the points of contact.
∠OAP = ∠OBP = 90° by the radius–tangent theorem.
OA = OB because they are radii of the same circle.
OP = OP because it is the common hypotenuse.
Therefore, △OAP ≅ △OBP by RHS congruence.
Hence, PA = PB by corresponding parts
of congruent triangles.
Circles Class 10 Formulas and Relationships
Let PA and PB be tangents from an external point P to a circle with centre O and radius r.
| Relationship | Result | Reason |
|---|---|---|
| Radius and tangent | OA ⟂ PA and OB ⟂ PB | Perpendicularity at the points of contact. |
| Tangent lengths | PA = PB | Tangents from the same external point. |
| Length of a tangent segment | PA = √(OP2 − r2) | Pythagoras’ theorem in △OAP. |
| Angle between tangents | ∠APB = 180° − ∠AOB | Angle sum of quadrilateral OAPB. |
| Angle bisector | ∠APO = ∠OPB | Congruence of △OAP and △OBP. |
Circles Class 10 Questions with Solutions
Example 1: Find a Tangent Length
Question: A circle has radius 5 cm. An external point P is 13 cm from its centre O. Find the length of a tangent PA.
OA ⟂ PA, so △OAP is right-angled at A.
PA2 = OP2 − OA2
PA2 = 132 − 52
PA2 = 144
PA = 12 cm
Example 2: Use Equal Tangents to Find x
Question: PA and PB are tangents from P to the same circle. If PA = 3x + 2 and PB = 5x − 6, find x and the tangent lengths.
PA = PB
3x + 2 = 5x − 6
8 = 2x
x = 4
PA = 3 × 4 + 2
PA = 14 units
Therefore, PB = 14 units.
Example 3: Find the Angle Between Two Tangents
Question: PA and PB touch a circle at A and B. If ∠AOB = 110°, find ∠APB.
∠OAP = 90°
∠OBP = 90°
∠APB + 110° + 90° + 90° = 360°
∠APB = 70°
Tangent Lengths in a Circumscribed Quadrilateral
A quadrilateral circumscribes a circle when all four sides touch the circle. Equal tangent segments from each vertex imply that the sums of opposite sides are equal.
Useful Result
If quadrilateral ABCD circumscribes a circle, then:
AB + CD = BC + AD.
Split each side at its point of contact. Pair the equal tangent segments from A, B, C and D, then add them. This explains the result rather than treating it as an unexplained formula.
Example 4: Find a Missing Side
Question: ABCD circumscribes a circle. If AB = 8 cm, BC = 6 cm and CD = 5 cm, find AD.
AB + CD = BC + AD
8 + 5 = 6 + AD
AD = 7 cm
How to Write Circles Class 10 Proofs
- Draw a labelled diagram showing the centre and contact points.
- Join the centre to each relevant point of contact.
- Mark the right angles formed by radii and tangents.
- Identify equal radii and equal tangent segments.
- Use congruence or angle relationships with reasons.
- Finish with the exact statement required by the question.
Do not assume tangent lengths are equal unless they originate from the same external point to the same circle. Similarly, a radius is perpendicular to a tangent specifically at the point of contact.
Circles Class 10 Mind Map and Revision 2026
Use the mind map of Circles Class 10 to connect tangents, perpendicular radii, equal lengths and congruent right triangles. Cover a branch and explain the theorem with its conditions before checking the notes.
For 2026 revision, practise direct length and angle questions before tackling longer proofs. Use the applicable NCERT exercises, commonly searched as Circles Class 10 Exercise 10.1 and Exercise 10.2, alongside suitable Exemplar and verified previous year questions.
Extra Questions and MCQs with Answers
-
A tangent is 8 cm long and the radius is 6 cm.
Find the distance from the external point to the centre.
Answer: 10 cm. -
Two tangents make an angle of 50°.
Find the corresponding smaller angle between the radii.
Answer: 130°. -
Two tangent lengths from the same external point are
4x − 1 and 2x + 5. Find x.
Answer: x = 3.
Circles Class 10 MCQ
1. How many tangents can be drawn from a point
outside a circle?
A. 0 B. 1 C. 2 D. Infinitely many
Answer: C.
2. The angle between a tangent and the radius
at its point of contact is:
A. 30° B. 45° C. 60° D. 90°
Answer: D.
Related Notes and Worksheet Practice
Apply the theorems using the Circles Class 10 Worksheet . Attempt each question independently and explain the theorem used in your working.
Revise proof-writing skills with Triangles Class 10 Notes and Mind Map , or explore the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this Circles Class 10 PDF contain notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for visual revision. Read the explanations first, then recall the theorem statements and connections through the map.
What is the difference between a tangent and a secant?
A tangent meets a circle at exactly one point. A secant intersects it at two distinct points. A chord is the segment joining two points on the circle, rather than the entire secant line.
Why is the radius perpendicular to the tangent?
The radius to the contact point is the shortest distance from the centre to the tangent line. Since the shortest distance from a point to a line is perpendicular, the radius and tangent form a right angle.
Why are two tangent lengths from an external point equal?
Join the centre to the contact points and the external point. The resulting right triangles have equal radii and a common hypotenuse. RHS congruence makes their corresponding tangent lengths equal.
Are all tangents to a circle equal?
No. The theorem concerns tangent segments drawn from the same external point. Segments originating from different external points need not have equal lengths.
Which side is the hypotenuse in a tangent-length question?
The segment joining the centre to the external point is the hypotenuse. The right angle is at the contact point, between the radius and tangent segment.
How do I find the angle between two tangents?
Subtract the smaller central angle between the radii to the contact points from 180°. This follows from the angle sum of the quadrilateral containing two right angles at the contact points.
Does the line joining the centre and external point bisect the tangent angle?
Yes. The two right triangles formed by the radii, tangent segments and common centre-to-point segment are congruent. Their angles at the external point are therefore equal.
Can a tangent pass through the centre of a circle?
No. A line through the centre intersects a circle at two points, so it is a secant. Its segment inside the circle is a diameter.
How can I improve at Circles Class 10 proof questions?
Practise joining the centre to contact points, identifying right triangles and writing reasons for equal lengths. State the congruence criterion before using corresponding parts.
Is Circles the same chapter as Areas Related to Circles?
No. Circles focuses on tangents and their geometrical properties. Areas Related to Circles focuses on circular areas, sectors, segments and related measurements. Revise each chapter according to its own concepts.
Where can I practise Circles Class 10 important questions?
Use the Circles Class 10 worksheet after revising the notes and mind map. Combine calculations with proof questions to develop both accuracy and reasoning.
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