Circles — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Circles (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Circles (CBSE & HBSE)
Circles focuses on tangent properties. CBSE tests the two tangent theorem and angle in semicircle. HBSE tests tangent-radius relationship and proof-based questions.
Tangent to a Circle
Key Definitions
| Term | Definition |
|---|---|
| Circle | Set of all points equidistant from a fixed point (centre) |
| Radius | Distance from centre to any point on circle |
| Chord | Line segment with both endpoints on circle |
| Secant | Line intersecting circle at TWO points |
| Tangent | Line touching circle at EXACTLY ONE point (point of tangency) |
Theorem 1: Tangent ⊥ Radius
Statement: The tangent at any point of a circle is perpendicular to the radius through that point.
Proof (indirect): Assume tangent PT is not perpendicular to radius OP. Then draw OM ⊥ PT where M is foot of perpendicular on PT. Since any other point M on PT is outside circle (tangent touches at only one point P), OM < OP (a perpendicular is shorter than any slant). But this contradicts the definition. Hence, OP ⊥ PT. ∎
Consequence: At point of tangency, tangent + radius form a 90° angle → enables use of Pythagoras!
Number of Tangents from a Point
| Position of Point | Number of Tangents |
|---|---|
| Inside the circle | 0 |
| On the circle | 1 (the tangent at that point) |
| Outside the circle | 2 |
Diagram Indicator: [Circle with centre O. External point P. Two tangents PA and PB touching circle at A and B. Right angles shown at A and B.]
Two Tangents from an External Point
Theorem 2: Equal Tangents
Statement: The lengths of the two tangents drawn from an external point to a circle are equal.
Proof: Let P be external point, PA and PB tangents to circle with centre O (tangent at A and B).
- In △OAP and △OBP:
- OA = OB (radii)
- OP = OP (common)
- ∠OAP = ∠OBP = 90° (tangent ⊥ radius)
- By RHS congruence: △OAP ≅ △OBP
- ∴ PA = PB ∎
Additional Results from Equal Tangents
- OP bisects angle APB (∠APO = ∠BPO)
- OP is the perpendicular bisector of AB
- ∠AOB + ∠APB = 180° (supplementary)
Using Equal Tangents in Problems
Example: A circle with centre O touches all four sides of quadrilateral ABCD. If AB = 6, BC = 7, CD = 4, find AD.
Using equal tangent theorem: if tangent touches at P (on AB), Q (on BC), R (on CD), S (on DA):
- AP = AS, BP = BQ, CQ = CR, DR = DS
- AB + CD = BC + DA (property of tangential quadrilateral)
- 6 + 4 = 7 + AD → AD = 3 cm
Tangential Quadrilateral Property
If a circle is inscribed in a quadrilateral ABCD: AB + CD = BC + DA
This property directly follows from the equal tangent theorem — essential for quadrilateral problems!
Angles in Semicircle and Chord Properties
Angle in a Semicircle
Theorem: The angle subtended by a diameter at any point on the circle is 90°.
If AB is diameter and C is any point on circle: ∠ACB = 90°
Angle at Centre vs Angle at Circumference
Theorem: Angle at centre = 2 × Angle at circumference (for same arc).
∠AOB = 2 × ∠ACB (where O is centre, C on major arc)
Cyclic Quadrilateral
Definition: A quadrilateral all of whose vertices lie on a circle.
Property: Opposite angles of a cyclic quadrilateral are supplementary (sum = 180°). ∠A + ∠C = 180° and ∠B + ∠D = 180°
Equal Chords and Their Properties
| Property | Statement |
|---|---|
| Equal chords | Equidistant from centre |
| Converse | Chords equidistant from centre are equal |
| Perpendicular from centre | Bisects the chord |
| Line joining centre to midpoint | Perpendicular to chord |
Tangent-Chord Angle (Alternate Segment Theorem)
Theorem: The angle between a tangent to a circle at a point and the chord drawn through that point is equal to the angle in the alternate segment.
∠(tangent, chord) = ∠ in alternate segment
CBSE Exam: The equal tangent theorem proof is a standard 5-mark question. Know every step and the RHS congruence.
Frequently asked questions
Are these Circles notes free?
Yes — the Circles notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Circles notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Circles chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Circles.