Circles Class 10 Notes: Tangent Theorems with Proofs and Solved Questions

Quick answer: Circles in Class 10 is a short chapter with two theorems, and one of them is asked as a proof almost every year. Learn both proofs and the five standard figures.

Definitions

• Tangent: a line touching the circle at exactly one point (point of contact).
• Secant: a line cutting the circle at two points.
• Exactly one tangent at any point on the circle; exactly two tangents from an external point.

Theorem 1

The tangent at any point is perpendicular to the radius through the point of contact. Proof: take any other point Q on the tangent; Q lies outside the circle, so OQ > OP for every Q. Hence OP is the shortest distance from O to the line → OP ⊥ tangent.

Theorem 2

Tangents from an external point are equal. Proof: in ΔOPA and ΔOPB, OA = OB (radii), OP common, ∠OAP = ∠OBP = 90° → congruent (RHS) → PA = PB. Also OP bisects ∠APB and ∠AOB.

Standard results

• Quadrilateral ABCD circumscribing a circle: AB + CD = AD + BC.
• Two parallel tangents: distance = diameter; the chord joining points of contact is a diameter.
• Angle between two tangents from P + angle at centre = 180°.

Solved questions

1. From a point 10 cm from the centre, tangent length to a circle of radius 6 cm: √(100 − 36) = 8 cm.

2. Tangents from P make 60° with each other → ∠AOB = 120°, and OP = 2r.

3. A circle touches the sides of a triangle (incircle) with sides 8, 10, 12: tangent lengths x, y, z with x + y = 8, y + z = 10, z + x = 12 → x = 5, y = 3, z = 7.

Common mistakes

• Assuming the tangent bisects the chord
• Using Pythagoras without marking the right angle at the point of contact
• Forgetting equal tangents when a circle is inside a quadrilateral

Related: Triangles Class 10.

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Frequently asked questions

Are these Circles Class 10 Notes notes enough for the exam?

They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.

How should I revise Circles Class 10 Notes?

Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.

What if I get stuck on a circles class 10 question?

Type or photograph the exact question on DoubtPay AI, choose your class and language, and get a step-by-step answer with a diagram. You can also ask for a hint or have your own attempt checked. Packs start at ₹9 for 15 answers.