Class 10 Maths Notes — Chapter 10: Circles
Tangents to a circle — length and angle properties.
Detailed NCERT notes
- A tangent to a circle is a line that touches the circle at exactly one point (the point of contact).
- Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Theorem: The lengths of tangents drawn from an external point to a circle are equal.
- From an external point P to a circle with centre O, if PA and PB are tangents at A and B, then PA = PB, OA ⟂ PA, OB ⟂ PB, and OP bisects ∠APB and ∠AOB.
- Quadrilateral OAPB has ∠OAP = ∠OBP = 90°, so ∠APB + ∠AOB = 180° (opposite angles supplementary).
- In a circle inscribed in a triangle (incircle), tangent lengths from each vertex are equal: if s = (a+b+c)/2, then AF = AE = s − a, BF = BD = s − b, CD = CE = s − c.
- For a quadrilateral circumscribing a circle, sum of opposite sides is equal: AB + CD = AD + BC.
- Construction problems: draw a tangent from an external point using the semicircle-on-diameter approach.
- Typical proofs use: right angle at radius, Pythagoras in right triangle, or supplementary angles in OAPB.
Formulas & key results
- Tangent ⟂ radius at point of contact
- Tangents from external point are equal in length
- In OAPB: ∠APB + ∠AOB = 180°
- Circumscribing quadrilateral: AB + CD = AD + BC
Mind map
- Tangent + radius = 90°
- External point → two equal tangents
- Cyclic quadrilateral OAPB
- Incircle in triangle: tangent lengths s − a etc.
Tricks & shortcuts
- Join centre to tangent point → right angle → Pythagoras.
- For quadrilateral circumscribing circle: opposite sides sum equal.
Common mistakes to avoid
- Assuming tangents from two different external points are equal.
- Confusing chord with tangent.
Competency-based questions & answers
- Q. PA and PB are tangents from P to a circle with centre O. If ∠APB = 60°, find ∠AOB.A. In OAPB, ∠AOB + 60° = 180° ⇒ ∠AOB = 120°.
- Q. Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller one.A. Chord ⟂ radius of inner at contact ⇒ half-chord = √(25 − 9) = 4 ⇒ chord = 8 cm.
- Q. A quadrilateral ABCD is drawn to circumscribe a circle. Prove AB + CD = AD + BC.A. Sum of tangent lengths from each vertex is equal, so grouping gives the identity.