CBSE Class 10 Maths: Circles — Tangents & Theorems Notes 2026

Introduction to Tangent

• Tangent: a line that touches a circle at exactly one point (point of tangency)
• Secant: line that intersects circle at two points; chord is a secant segment
• A tangent is perpendicular to the radius drawn to the point of contact

Theorem: Tangent Perpendicular to Radius

• Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact
• Proof: by contradiction using distance property
• Corollary: at any point on a circle, only one tangent can be drawn

Tangents from an External Point

• Two tangents can be drawn from an external point to a circle
• Theorem: lengths of tangents from an external point are equal
• Proof: using congruent triangles (RHS congruence: radius equal, right angle, hypotenuse equal)

Angle Between Tangents

• If two tangents from external point T touch circle at A and B: OA⊥TA, OB⊥TB
• Quadrilateral OAPB: sum of angles = 360°; ∠AOB + ∠ATB = 180°
• ∠OAT = ∠OBT = 90°; OA = OB (radii); so triangles OAT ≅ OBT (RHS)

Circles and Quadrilaterals

• Tangent to circle from vertices of quadrilateral: if circle touches all four sides, AB+CD = BC+AD
• Proof using equal tangent lengths from each vertex
• CBSE theorem: tangents drawn from external point are equal — used in quadrilateral proofs

Constructions — Tangents

• Draw tangent at a point on the circle: draw radius, construct perpendicular
• Draw two tangents from external point: bisect OP (O=centre, P=external point) to find midpoint M; draw circle with radius OM; intersections with original circle are tangent points
• CBSE construction carries 3 marks; use correct instruments

CBSE Exam Questions — Circles

• Prove: tangent lengths from external point are equal (standard CBSE proof)
• If tangents from P to circle with centre O are PA and PB, and angle APB = 70°, find angle AOB
• Quadrilateral ABCD circumscribes circle; prove AB + CD = AD + BC