Class 10 Maths Chapter 10 Circles MCQ / Objective Questions with Answers

Contents hide
1 Direct Questions
2 Assertion-Reason
3 Match the Following
4 Statement-Based Questions
5 Fill in the Blanks
6 Odd One Out

19 multiple-choice (MCQ / objective) questions from this chapter, in the common exam-style shapes. Every question comes with an explanation along with the answer, so you remember why the option is correct, not just which one it is.

1 Direct Questions

Direct Questions
  1. 1 markA tangent to a circle intersects the circle in how many points?
    (A) two(B) exactly one(C) none(D) infinitely many
  2. 1 markThe length of the tangent from a point at distance 10 cm from the centre of a circle of radius 6 cm is
    (A) 66 cm(B) 1212 cm(C) 88 cm(D) 1616 cm
  3. 1 markTP and TQ are tangents from T to a circle with centre O. If ∠POQ=100∘\angle POQ = 100^\circ, then ∠PTQ\angle PTQ is
    (A) 80∘80^\circ(B) 50∘50^\circ(C) 90∘90^\circ(D) 100∘100^\circ
  4. 1 markTwo concentric circles have radii 13 cm and 5 cm. The length of the chord of the larger circle which touches the smaller circle is
    (A) 2424 cm(B) 1212 cm(C) 1818 cm(D) 2626 cm
  5. 1 markA quadrilateral ABCD circumscribes a circle. If AB = 6 cm, BC = 7 cm and CD = 4 cm, then AD is
    (A) 55 cm(B) 1717 cm(C) 33 cm(D) 99 cm
  6. 1 markA parallelogram circumscribes a circle and one of its sides is 5 cm. Its perimeter is
    (A) 1010 cm(B) 2020 cm(C) 1515 cm(D) cannot be found
  7. 1 markA circle is inscribed in △ABC\triangle ABC, touching BC at D, CA at E and AB at F. If AF = 3 cm, BD = 4 cm and CE = 5 cm, the perimeter of △ABC\triangle ABC is
    (A) 1212 cm(B) 2424 cm(C) 1818 cm(D) 3030 cm
  8. 1 markIn Example 3 of the book, PQ is a chord of length 8 cm of a circle of radius 5 cm and the tangents at P and Q meet at T. TP equals
    (A) 163\dfrac{16}{3} cm(B) 44 cm(C) 253\dfrac{25}{3} cm(D) 203\dfrac{20}{3} cm
  9. 1 markTangents PA and PB from P to a circle with centre O and radius 3 cm are perpendicular to each other. OP is
    (A) 66 cm(B) 323\sqrt{2} cm(C) 33 cm(D) 99 cm

2 Assertion-Reason

Assertion-Reason
  1. 1 markAssertion (A): The length of the tangent from a point 13 cm away from the centre of a circle of radius 5 cm is 12 cm.
    Reason (R): The tangent at any point of a circle is perpendicular to the radius through the point of contact.
    (A) Both A and R are true, and R is the correct explanation of A(B) Both A and R are true, but R is not the correct explanation of A(C) A is true, but R is false(D) A is false, but R is true
  2. 1 markAssertion (A): A tangent can be drawn to a circle through a point lying inside the circle.
    Reason (R): Every line through a point inside a circle cuts the circle in two points.
    (A) Both A and R are true, and R is the correct explanation of A(B) Both A and R are true, but R is not the correct explanation of A(C) A is true, but R is false(D) A is false, but R is true
  3. 1 markAssertion (A): If the radii to the points of contact of two tangents make an angle of 110∘110^\circ at the centre, the angle between the tangents is 70∘70^\circ.
    Reason (R): The angle between two tangents from an external point is equal to the angle subtended at the centre by the line joining the points of contact.
    (A) Both A and R are true, and R is the correct explanation of A(B) Both A and R are true, but R is not the correct explanation of A(C) A is true, but R is false(D) A is false, but R is true

3 Match the Following

Match the Following
  1. 1 markMatch Column I (term) with Column II (meaning).
    Column I: 1) Tangent   2) Secant   3) Point of contact   4) Normal
    Column II: i. line containing the radius through the point of contact   ii. line meeting the circle in exactly one point   iii. common point of a tangent and the circle   iv. line meeting the circle in two points
    (A) 1-iv, 2-ii, 3-iii, 4-i(B) 1-ii, 2-iv, 3-i, 4-iii(C) 1-ii, 2-iv, 3-iii, 4-i(D) 1-iii, 2-iv, 3-ii, 4-i
  2. 1 markA circle has radius rr and a point is at distance dd from its centre. Match each pair with the length of the tangent.
    Column I: 1) r=3,d=5r = 3, d = 5   2) r=5,d=13r = 5, d = 13   3) r=8,d=17r = 8, d = 17   4) r=7,d=25r = 7, d = 25 (all in cm)
    Column II: i. 15 cm   ii. 24 cm   iii. 4 cm   iv. 12 cm
    (A) 1-iii, 2-iv, 3-i, 4-ii(B) 1-iv, 2-iii, 3-i, 4-ii(C) 1-iii, 2-iv, 3-ii, 4-i(D) 1-i, 2-iv, 3-iii, 4-ii

4 Statement-Based Questions

Statement-Based Questions
  1. 1 markStatement I: The tangent at a point of a circle is parallel to the radius through that point.
    Statement II: The length of the tangent from an external point is the length of the tangent segment from that point to the point of contact.
    (A) Both Statement I and Statement II are true(B) Statement I is true, but Statement II is false(C) Statement I is false, but Statement II is true(D) Both Statement I and Statement II are false
  2. 1 markStatement I: A circle can have at most one tangent parallel to a given line.
    Statement II: The tangents drawn at the two ends of a diameter of a circle intersect each other.
    (A) Both Statement I and Statement II are true(B) Statement I is true, but Statement II is false(C) Statement I is false, but Statement II is true(D) Both Statement I and Statement II are false

5 Fill in the Blanks

Fill in the Blanks
  1. 1 markThe common point of a tangent to a circle and the circle is called the ______.
    (A) centre(B) point of contact(C) point of intersection of secants(D) normal
  2. 1 markIn the proof of Theorem 10.2, △OQP≅△ORP\triangle OQP \cong \triangle ORP by the ______ congruence rule.
    (A) RHS(B) SAS(C) ASA(D) SSS

6 Odd One Out

Odd One Out
  1. 1 markPA and PB are tangents from an external point P to a circle with centre O. Which of these is the odd one out, being NOT always true?
    (A) PA=PBPA = PB(B) ∠OAP=90∘\angle OAP = 90^\circ(C) OP bisects ∠APB\angle APB(D) ∠APB=∠AOB\angle APB = \angle AOB
Answer Key and Explanations
Direct Questions
  1. (B) By definition a tangent meets the circle in only one point, called the point of contact.
  2. (C) Radius ⊥\perp tangent (Theorem 10.1), so l2=d2−r2=102−62=100−36=64l^2 = d^2 – r^2 = 10^2 – 6^2 = 100 – 36 = 64, giving l=8l = 8 cm.
  3. (A) ∠OPT=∠OQT=90∘\angle OPT = \angle OQT = 90^\circ, and the angles of quadrilateral OPTQ add to 360∘360^\circ: ∠PTQ=360∘−180∘−100∘=80∘\angle PTQ = 360^\circ – 180^\circ – 100^\circ = 80^\circ.
  4. (A) The radius 5 cm to the point of contact is ⊥\perp the chord and bisects it. Half-chord =132−52=144=12= \sqrt{13^2 – 5^2} = \sqrt{144} = 12 cm, so the chord =24= 24 cm.
  5. (C) AB+CD=AD+BCAB + CD = AD + BC, so 6+4=AD+76 + 4 = AD + 7, giving AD=3AD = 3 cm.
  6. (B) A parallelogram circumscribing a circle is a rhombus, so all four sides are 5 cm: perimeter =4×5=20= 4 \times 5 = 20 cm.
  7. (B) Equal tangents: AE = AF = 3, BF = BD = 4, CD = CE = 5. Perimeter =2(3+4+5)=24= 2(3 + 4 + 5) = 24 cm.
  8. (D) OR = 3 cm, and △TRP∼△PRO\triangle TRP \sim \triangle PRO gives TP5=43\dfrac{TP}{5} = \dfrac{4}{3}, so TP=203TP = \dfrac{20}{3} cm. (163\dfrac{16}{3} is TR and 253\dfrac{25}{3} is OT.)
  9. (B) ∠OAP=∠OBP=90∘\angle OAP = \angle OBP = 90^\circ (Theorem 10.1) and ∠APB=90∘\angle APB = 90^\circ, so OAPB has right angles at A, P and B and is a rectangle; since OA = OB = 3 cm it is a square of side 3 cm, and OP is its diagonal: OP=32+32=32OP = \sqrt{3^2 + 3^2} = 3\sqrt{2} cm.
Assertion-Reason
  1. (A) R gives a right angle at the point of contact, so l=132−52=12l = \sqrt{13^2 – 5^2} = 12 cm. Both are true and R explains A.
  2. (D) A is false (Case 1: there is no tangent through an inside point). R is true, and it is the reason A is false.
  3. (C) A is true: 180∘−110∘=70∘180^\circ – 110^\circ = 70^\circ. R is false: the two angles are supplementary (they add to 180∘180^\circ), not equal.
Match the Following
  1. (C) Tangent: one point (ii). Secant: two points (iv). Point of contact: common point (iii). Normal: line of the radius through the point of contact (i).
  2. (A) l=d2−r2l = \sqrt{d^2 – r^2}: 25−9=4\sqrt{25-9} = 4, 169−25=12\sqrt{169-25} = 12, 289−64=15\sqrt{289-64} = 15, 625−49=24\sqrt{625-49} = 24.
Statement-Based Questions
  1. (C) I is false: the tangent is perpendicular to that radius (Theorem 10.1). II is the book’s definition of the length of a tangent, so it is true.
  2. (D) I is false: there can be two such tangents. II is false: tangents at the ends of a diameter are parallel, so they never meet.
Fill in the Blanks
  1. (B) NCERT Ex 10.1 Q2(iv): it is called the point of contact.
  2. (A) The triangles are right-angled at Q and R, share the hypotenuse OP and have equal radii OQ = OR: Right angle, Hypotenuse, Side.
Odd One Out
  1. (D) PA = PB (Thm 10.2), ∠OAP=90∘\angle OAP = 90^\circ (Thm 10.1) and OP bisects ∠APB\angle APB (Remark 2) are always true. ∠APB\angle APB and ∠AOB\angle AOB are supplementary, not equal.
More for this chapter
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