ICSE Class 10 Mathematics: Circles — Important Questions with Answers 2026

Short Answer Questions (2-3 Marks)

• Q: Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
  Ans: Let O be centre, P be point of contact, PT be tangent. Assume OP is not perpendicular to PT. Then there exists a point Q on PT such that OQ ⊥ PT. OQ < OP (perpendicular is shortest distance). But Q is on the tangent (outside circle), so OQ > radius = OP. Contradiction. Hence OP ⊥ PT.
• Q: From an external point, two tangents are drawn to a circle. Prove that they are equal in length.
  Ans: Let PA and PB be tangents from external point P to circle with centre O, touching at A and B. In △OAP and △OBP: OA = OB (radii), OP = OP (common), ∠OAP = ∠OBP = 90° (radius ⊥ tangent). By RHS congruence: △OAP ≅ △OBP. Therefore PA = PB.
• Q: In the given figure, AB is a diameter and CD is a chord. If AB = 10 cm and CD = 8 cm, find the distance of CD from the centre.
  Ans: OA = OB = 5 cm (radius). Let OM ⊥ CD. Then CM = MD = 8/2 = 4 cm (perpendicular from centre bisects chord). In right △OMC: OM² = OC² - CM² = 5² - 4² = 25 - 16 = 9. OM = 3 cm.

Long Answer / Application Questions (4-6 Marks)

• Q: Prove that the angle in a semicircle is a right angle.
  Ans: Let AB be diameter, C be any point on the circle. ∠ACB is the angle in the semicircle. Arc ADB (major arc) subtends ∠AOB = 180° at centre (straight line). Angle subtended at centre = 2 × angle at circumference. So ∠ACB = 180°/2 = 90°. Hence angle in a semicircle is a right angle.

Exam Tips for This Chapter

• Revise all definitions and laws from Circles — they are commonly asked as 1-2 mark questions
• Practice diagrams related to Circles — neat labelled diagrams carry 2-3 marks
• For numericals, always show formula → substitution → answer with correct units
• Previous year analysis shows Circles carries 8-12 marks in the board exam