Circles Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 10 notes: tangents to a circle, the two key tangent theorems, and the number of tangents from a point, with worked proofs and examples.
Circles — Class 10 Maths Notes
Chapter Snapshot
This chapter is about tangents to a circle. You learn what a tangent is, the two key theorems — a tangent is perpendicular to the radius at the point of contact, and tangents from an external point are equal — and how many tangents can be drawn from a point.
Board relevance: a compact Geometry chapter that reliably gives one tangent-theorem proof and one numerical using the equal-tangents property. Draw a clear figure and justify each step.
Key Concepts & Definitions
Secant — a line that intersects a circle at two points.
Tangent — a line that touches the circle at exactly one point, called the point of contact. A tangent is the limiting position of a secant when its two intersection points merge into one.
Number of tangents from a point:
Position of point Number of tangents
Inside the circle 0 (no tangent possible)
On the circle 1 (exactly one)
Outside the circle 2
Formulas — The Two Theorems
Theorem 1 — Tangent ⟂ Radius
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
If a tangent touches a circle (centre O) at P, then OP ⟂ tangent, i.e. the angle between the radius OP and the tangent at P is 90°.
Consequence: OP is the shortest distance from the centre to the tangent line.
Theorem 2 — Equal tangents from an external point
The lengths of the two tangents drawn from an external point to a circle are equal.
From an external point T, if TA and TB are tangents touching the circle at A and B, then TA = TB.
Proof idea: In triangles OAT and OBT, OA = OB (radii), OT is common, and ∠OAT = ∠OBT = 90° (Theorem 1). By RHS congruence, the triangles are congruent, so TA = TB. Also OT bisects ∠ATB and ∠AOB.
Worked Examples
Example 1 — Length of a tangent: A tangent from a point T at distance 13 cm from the centre O touches a circle of radius 5 cm at P. Find TP.
OP ⟂ TP (Theorem 1), so △OPT is right-angled at P. TP = √(OT² − OP²) = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.
Example 2 — Equal tangents: From an external point, two tangents of length 8 cm are drawn. If they make an angle of 60° with each other at the external point, describe the figure.
By Theorem 2 both tangents are 8 cm. Since OT bisects the 60° angle, each half is 30°, and the two right triangles OAT, OBT are congruent — a standard set-up for finding the radius using tan 30° = radius/8.
Example 3 — Quadrilateral property: A circle touches all four sides of quadrilateral ABCD at P, Q, R, S. Prove AB + CD = AD + BC.
Using equal tangents from each vertex (AP = AS, BP = BQ, CR = CQ, DR = DS) and adding suitably gives AB + CD = AD + BC. This is a favourite proof using Theorem 2 repeatedly.
Example 4 — Angle between tangents: Two tangents from external point T touch a circle at A and B, with ∠AOB = 110°. Find ∠ATB.
OATB is a quadrilateral with ∠OAT = ∠OBT = 90°. Angle sum = 360°: ∠ATB = 360° − 90° − 90° − 110° = 70°. (Note ∠AOB + ∠ATB = 180° — the tangents' angle and the central angle are supplementary.)
Important Question Patterns
1. Length of tangent (2–3 marks): use OP ⟂ tangent → right triangle → Pythagoras to find the tangent length or radius.
2. Prove equal tangents (3 marks): the standard RHS-congruence proof of Theorem 2.
3. Angle problems (3 marks): angle between two tangents and the central angle are supplementary (∠ATB + ∠AOB = 180°); use the quadrilateral angle sum.
4. Tangent-and-quadrilateral (3–4 marks): a circle inscribed in a quadrilateral → AB + CD = AD + BC using equal tangents.
5. Two-circle / touching figures (3–4 marks): apply the two theorems to find lengths where tangents and radii meet.
⚡ Quick Revision
- Tangent touches the circle at one point (point of contact); a secant cuts it at two.
- Tangents from a point: 0 inside, 1 on, 2 outside the circle.
- Theorem 1: tangent ⟂ radius at the point of contact (angle = 90°) → gives a right triangle for Pythagoras.
- Theorem 2: tangents from an external point are equal (TA = TB); proved by RHS congruence; OT bisects the angle.
- Length of tangent from external point: √(d² − r²) where d = distance to centre, r = radius.
- Angle between two tangents + central angle = 180° (supplementary).
- Inscribed quadrilateral: AB + CD = AD + BC.
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