Class 10 Mathematics: Circles important questions

Circles is a chapter of Class 10 Mathematics for the Andhra Pradesh State Board. These 15 questions are marked Very Important, Important or Practice, and each answer has been checked by our team. Use them to revise, not as a prediction of what will come in the exam.

1. Practice · 1 marks

   1. How many common tangents can be drawn to two concentric circles?

   None (0). Any line touching the inner circle cuts the outer circle, so no line can touch both.

2. Practice · 1 marks

   2. How many tangents can a circle have?

   Infinitely many: there is exactly one tangent at each point of the circle.

3. Important · 2 marks

   3. Find the length of the tangent from a point 15 cm from the centre of a circle of radius 9 cm.

   The tangent is perpendicular to the radius at the point of contact, so length = √(15² − 9²) = √(225 − 81) = √144 = 12 cm.

4. Important · 4 marks

   4. Prove that the perpendicular at the point of contact to the tangent of a circle passes through the centre.

   Let the tangent at P touch the circle with centre O. The radius OP is perpendicular to the tangent at P (a tangent is perpendicular to the radius through the point of contact). At the point P there is only one line perpendicular to the tangent. Since OP is perpendicular to the tangent at P, that perpendicular must be the line OP. So the perpendicular at the point of contact passes through the centre O.

5. Practice · 1 marks

   5. Create a geometrical design using two tangents drawn from an external point to a circle.

   Draw a circle with centre O and a point P outside it. Draw the two tangents PA and PB touching the circle at A and B. The figure looks like an ice-cream cone: the circle sits in the 'V' formed by PA and PB, with PA = PB and OA ⟂ PA, OB ⟂ PB.

6. Important · 2 marks

   6. A quadrilateral ABCD is drawn to circumscribe a circle. Write the relation between its sides.

   AB + CD = AD + BC. (Tangents from each vertex to the circle are equal, so the sums of opposite sides are equal.)

7. Important · 4 marks

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

   Let parallelogram ABCD touch the circle at P (on AB), Q (on BC), R (on CD) and S (on DA). Tangents from a point are equal: AP = AS, BP = BQ, CR = CQ, DR = DS. Adding: (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ), so AB + CD = AD + BC. In a parallelogram AB = CD and AD = BC, so 2AB = 2AD and AB = AD. So all four sides are equal, and ABCD is a rhombus.

8. Practice · 1 marks

   8. Draw a rough sketch showing a tangent to a circle passing through the end point of its radius.

   Draw a circle with centre O and a radius OP. At P, draw a line perpendicular to OP. That line touches the circle only at P: it is the tangent at P, and OP ⟂ tangent.

9. Important · 4 marks

   9. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.

   Let ∠PTQ = θ. TP = TQ (tangents from an external point), so ΔTPQ is isosceles and ∠TPQ = ∠TQP = (180° − θ)/2 = 90° − θ/2. ∠OPT = 90° (radius ⟂ tangent), so ∠OPQ = ∠OPT − ∠TPQ = 90° − (90° − θ/2) = θ/2. Hence ∠PTQ = θ = 2∠OPQ.

10. Practice · 1 marks

    10. Draw a rough sketch to represent this: the chord of a larger circle is a tangent to a smaller concentric circle.

    Draw two circles with the same centre O, one small and one large. Draw a chord AB of the large circle that just touches the small circle at one point P. The radius OP of the small circle is perpendicular to AB at P, and P is the midpoint of AB.

11. Important · 2 marks

    11. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at Q so that OQ = 13 cm. Find the length of PQ.

    The tangent is perpendicular to the radius, so ∠OPQ = 90°. PQ² = OQ² − OP² = 13² − 5² = 169 − 25 = 144, so PQ = 12 cm.

12. Important · 4 marks

    12. Joffer thinks that the lengths of two tangents drawn to a circle from an external point are not always equal. How would you resolve his conflict?

    Show him the theorem: tangents drawn from an external point to a circle are equal. Proof: Let P be the external point, O the centre, and PA, PB the tangents touching at A and B. Join OA, OB and OP. In ΔOAP and ΔOBP: OA = OB (radii), OP = OP (common), ∠OAP = ∠OBP = 90° (radius ⟂ tangent). So ΔOAP ≅ ΔOBP (RHS), and PA = PB. The two tangents are always equal.

13. Practice · 1 marks

    13. Draw a rough sketch of a circle and two lines parallel to a given line such that one is a tangent and the other is a secant to the circle.

    Draw a circle and a line l outside it. Draw a line parallel to l that just touches the circle at one point: this is the tangent. Draw another line parallel to l that cuts the circle at two points: this is the secant.

14. Important · 2 marks

    14. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at Q so that OQ = 12 cm. Find the length of PQ.

    OP ⟂ PQ, so PQ² = OQ² − OP² = 144 − 25 = 119. PQ = √119 cm ≈ 10.9 cm.

15. Important · 4 marks

    15. A circular rainwater-harvesting pit has radius 4 m. Two support wires are drawn from a common external point P and touch the pit boundary at A and B. a) Draw a suitable figure. b) What can you say about the lengths of the two wires? c) Name the theorem used. d) What angle does a radius make with a wire at the point of contact?

    a) Draw a circle with centre O and radius 4 m. Mark a point P outside it and draw the two tangents PA and PB touching the circle at A and B. Join OA and OB. b) The wires are equal: PA = PB. c) The lengths of tangents drawn from an external point to a circle are equal. d) 90°: the radius is perpendicular to the tangent at the point of contact.