Keerthana Classes

Maths ch10 circles

October 04, 2026

37 marks. Read the notes, then tap Start test. Answers and your score appear only after you submit.

Quick notes

A tangent meets a circle at exactly one point (the point of contact). A secant cuts it at two points.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

From a point inside the circle there is no tangent; from a point on the circle there is exactly one; from a point outside there are exactly two.

The lengths of the two tangents drawn from an external point are equal.

Part A: Multiple choice 16 questions

1. A tangent meets a circle at

  1. no point
  2. two points
  3. one point
  4. infinitely many points

That is the definition of a tangent.

2. The number of tangents a circle can have is

  1. 1
  2. 2
  3. 0
  4. infinitely many

There is one tangent at every point of the circle.

3. The tangent at any point of a circle is ___ to the radius through the point of contact.

  1. perpendicular
  2. parallel
  3. equal
  4. inclined at 45°

This is Theorem 10.1.

4. The number of tangents from a point inside a circle is

  1. 1
  2. 0
  3. 2
  4. infinitely many

Every line through an inside point cuts the circle twice.

5. The number of tangents from a point outside a circle is

  1. 1
  2. 0
  3. 2
  4. 3

Exactly two tangents can be drawn.

6. The lengths of tangents from an external point to a circle are

  1. unequal
  2. in ratio 1 : 2
  3. perpendicular
  4. equal

This is Theorem 10.2.

7. A tangent from Q is 24 cm long and Q is 25 cm from the centre. The radius is

  1. 7 cm
  2. 12 cm
  3. 15 cm
  4. 24.5 cm

r = √(625 − 576) = 7 cm.

8. A line that cuts a circle at two points is a

  1. tangent
  2. secant
  3. radius
  4. diameter

A secant has two common points with the circle.

9. How many tangents of a circle can be parallel to a given secant?

  1. 1
  2. 0
  3. 2
  4. infinitely many

One on each side of the circle.

10. TP and TQ are tangents and ∠POQ = 110°. ∠PTQ is

  1. 110°
  2. 90°
  3. 80°
  4. 70°

∠PTQ = 180° − 110° = 70°.

11. Tangents PA and PB are inclined at 80°. ∠POA is

  1. 50°
  2. 40°
  3. 60°
  4. 70°

∠AOB = 100°, and OP bisects it.

12. The radius is 5 cm and the tangent from P is 12 cm. OP is

  1. 17 cm
  2. 13 cm
  3. 7 cm
  4. 10 cm

OP = √(25 + 144) = 13 cm.

13. Two concentric circles have radii 5 cm and 3 cm. A chord of the larger circle touching the smaller one is

  1. 4 cm
  2. 6 cm
  3. 8 cm
  4. 10 cm

Half chord = √(25 − 9) = 4, so the chord is 8 cm.

14. If quadrilateral ABCD circumscribes a circle, then

  1. AB = CD
  2. AB + BC = CD + DA
  3. AC = BD
  4. AB + CD = AD + BC

Using equal tangent lengths from each vertex.

15. A parallelogram that circumscribes a circle is a

  1. rhombus
  2. rectangle
  3. trapezium
  4. kite

AB + CD = AD + BC gives AB = AD.

16. The tangents at the two ends of a diameter are

  1. perpendicular
  2. parallel
  3. equal to the diameter
  4. intersecting

Both are perpendicular to the same diameter.

Part B: Fill in the blanks 8 blanks

17. A line touching a circle at exactly one point is a [[tangent]].
18. The common point of a tangent and the circle is called the point of [[contact]].
19. The tangent at a point is [[perpendicular]] to the radius through that point.
20. From a point on the circle, exactly [[1|one]] tangent can be drawn.
21. From a point outside the circle, [[2|two]] tangents can be drawn.
22. If OP = 10 cm and the radius is 6 cm, the tangent from P is [[8]] cm long.
23. A circle has [[infinitely many|infinite|infinitely]] tangents.
24. The tangents from an external point to a circle are [[equal]] in length.

Part C: Questions and answers 9 questions

25. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Let XY touch the circle at P. Take any other point Q on XY. Q lies outside the circle, so OQ > OP.
So OP is the shortest distance from O to XY, and the shortest distance is the perpendicular. Hence OP ⟂ XY.

26. Prove that the lengths of tangents drawn from an external point to a circle are equal.

Let PQ and PR be tangents from P. In ΔOQP and ΔORP: OQ = OR (radii), OP is common, ∠OQP = ∠ORP = 90°.
So ΔOQP ≅ ΔORP (RHS), and PQ = PR.

27. A point A is 5 cm from the centre of a circle, and the tangent from A is 4 cm. Find the radius.

The radius is perpendicular to the tangent: r² + 4² = 5².
r = 3 cm.

28. Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller circle.

The radius of the small circle is perpendicular to the chord and bisects it.
Half chord = √(5² − 3²) = 4, so chord = 8 cm.

29. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

Let the sides touch at P, Q, R, S. Tangents from each vertex are equal: AP = AS, BP = BQ, CR = CQ, DR = DS.
Adding: AB + CD = AS + BQ + CQ + DS = AD + BC.

30. Prove that a parallelogram circumscribing a circle is a rhombus.

From the previous result, AB + CD = AD + BC. In a parallelogram AB = CD and AD = BC, so 2AB = 2AD and AB = AD.
All sides are equal, so it is a rhombus.

31. TP and TQ are tangents from T to a circle with centre O. Prove that ∠PTQ = 2∠OPQ.

Let ∠PTQ = θ. TP = TQ, so ∠TPQ = ∠TQP = (180° − θ)/2 = 90° − θ/2.
∠OPT = 90°, so ∠OPQ = 90° − (90° − θ/2) = θ/2. Hence ∠PTQ = 2∠OPQ.

32. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Each tangent is perpendicular to the diameter at its end point. Two lines perpendicular to the same line are parallel.

33. Prove that the angle between two tangents from an external point is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

In quadrilateral OAPB: ∠OAP = ∠OBP = 90°. The angles add up to 360°, so ∠APB + ∠AOB = 180°.

Part D: Word puzzles 4 puzzles

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