CBSE Class 10 Mathematics Circles Worksheet Set B

Question. PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that ∠POR = 120°, then ∠OPQ is
(a) 60°
(b) 45°
(c) 30°
(d) 90°

Answer: C

Question. If a regular hexagon is inscribed in a circle of radius r, then its perimeter is
(a) 3r
(b) 6r
(c) 9r
(d) 12r

Answer: B

Question. AB and CD are two chords of a circle intersecting at the point P outside the circle. If PA = 12 cm, CD = 7cm and PC = 15 cm, then AB is equal to
(a) 15.5 cm
(b) 4 cm
(c) 8 cm
(d) 10 cm

Answer: A

Question. In two concentric circles, if chords are drawn in the outer circle which touch the inner circle, then
(a) all chords are of different lengths.
(b) all chords are of same length.
(c) only parallel chords are of same length.
(d) only perpendicular chords are of same length.

Answer: B

Question. Number of tangents to a circle which are parallel to a secant, is
(a) 3
(b) 2
(c) 1
(d) infinite

Answer: B

Question. AB and CD are two common tangents to circles which touch each other at a point C. If D lies on AB such that CD = 4 cm, then AB is
(a) 12 cm
(b) 8 cm
(c) 4 cm
(d) 6 cm

Answer: B

Question. Two circles, both of radii a touch each other and each of them touches internally a circle of radius 2a. Then the radius of the circle which touches all the three circles is
(a) (1/2)a
(b) (2/3)a
(c) (3/4)a
(d) a

Answer: B

Question. Let ABCD be a square of side length 1, and G a circle passing through B and C, and touching AD. The radius of G is
(a) 3/8
(b) 1/2
(c) 1/√2
(d) 5/8

Answer: D

Question. Three circles of radii 1, 2 and 3 units respectively touch each other externally in the plane. The circumradius of the triangle formed by joining the centers of the circles is
(a) 1.5
(b) 2
(c) 2.5
(d) 3

Answer: C

Question. The length of tangent drawn from a point Q to a circle is 24 cm and distance of Q from the centre of circle is 25 cm. The radius of circle is
(a) 7 cm
(b) 12 cm
(c) 15 cm
(d) 24.5 cm

Answer: A

Question. Which of the following is a cyclic quadrilateral?
(a) Rhombus
(b) Rectangle
(c) Parallelogram
(d) Trapezium

Answer: B

Question. Which of the following is/are not correct?
(a) A secant is a line that intersects a circle in two distinct points.
(b) In a circle, the perpendicular from the centre to a chord bisects the chord.
(c) The point common to a circle and its tangent is called the point of contact.
(d) Adjacent angles of a cyclic quadrilateral are supplementary.

Answer: D

Question. Which of the following statement(s) is / are not correct ?
(a) The length of tangent from an external point P on circle with centre O is always less than OP.
(b) The tangent to the circumcircle of an isosceles triangle ABC at A, in which AB = AC, is parallel to BC.
(c) If angle between two tangents drawn from a point P to a circle of radius ‘a’ and centre ‘O’is 90°, then OP = a√2.
(d) None of these

Answer: D

Question. Which of the following statement(s) is/are correct?
(a) If a chord AB subtends an angle of 60° at the centre of a circle, then angle between the tangents at A and B is also 60°.
(b) The length of tangent from an external point on a circle is always greater than the radius of the circle.
(c) If a number of circle touch a given line segment PQ at a point A, then their centres lie on the perpendicular bisector of PQ.
(d) None of these

Answer: D

Question. Which of the following statement(s) is/are incorrect?
(a) Angle between the tangent line and the radius at the point of contact is 90°.
(b) A circle can have two parallel tangents atmost.
(c) The distance between two parallel tangents drawn to a circle is equal to twice of radius.
(d) A line intersecting a circle in two points is called a chord.

Answer: D

Question. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(a) 12 cm
(b) 13 cm
(c) 8.5 cm
(d) √119 cm

Answer: D

Question. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80°, then ∠POA is equal to
(a) 50°
(b) 60°
(c) 70°
(d) 80°

Answer: A

Question. If angle between two radii of a circle is 130°, the angle between the tangents at the ends of the radii is :
(a) 90°
(b) 50°
(c) 70°
(d) 40°

Answer: B

DIRECTIONS : Study the given Case/Passage and answer the following questions.

Case/Passage-I

A Ferris wheel (or a big wheel in the United Kingdom) is an amusement ride consisting of a rotating upright wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, capsules, gondolas, or pods) attached to the rim in such a way that as the wheel turns, they are kept upright, usually by gravity. After taking a ride in Ferris wheel, Aarti came out from the crowd and was observing her friends who were enjoying the ride . She was curious about the different angles and measures that the wheel will form. She forms the figure as given below.

Question. In the given figure find ∠ROQ
(a) 60
(b) 100
(c) 150
(d) 90

Answer: C

Question. Find ∠RQP
(a) 75
(b) 60
(c) 30
(d) 90

Answer: A

Question. Find ∠RSQ
(a) 60
(b) 75
(c) 100
(d) 30

Answer: B

Question. Find ∠ORP
(a) 90
(b) 70
(c) 100
(d) 60

Answer: A

Case/Passage-II

Varun has been selected by his School to design logo for Sports Day T-shirts for students and staff. The logo design is as given in the figure and he is working on the fonts and different colours according to the theme.

In given figure, a circle with centre O is inscribed in a ΔABC, such that it touches the sides AB, BC and CA at points D, E and F respectively. The lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm respectively.

Question. Find the length of AD
(a) 7
(b) 8
(c) 5
(d) 9

Answer: A

Question. Find the Length of BE
(a) 8
(b) 5
(c) 2
(d) 9

Answer: B

Question. Find the length of CF
(a) 9
(b) 5
(c) 2
(d) 3

Answer: D

Question. If radius of the circle is 4cm, Find the area of ΔOAB
(a) 20
(b) 36
(c) 24
(d) 48

Answer: C

Question. Find area of ΔABC
(a) 50
(b) 60
(c) 100
(d) 90

Answer: B

Assertion & Reason

DIRECTIONS : Each of these questions contains an Assertion followed by Reason. Read them carefully and answer the question on the basis of following options. You have to select the one that best describes the two statements.
(a) If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
(b) If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
(c) If Assertion is correct but Reason is incorrect.
(d) If Assertion is incorrect but Reason is correct.

Question. Assertion: If in a circle, the radius of the circle is 3 cm and distance of a point from the centre of a circle is 5 cm, then length of the tangent will be 4 cm.
Reason:(hypotenuse)² = (base)² + (height)²
Answer: A

Question. Assertion: If in a cyclic quadrilateral, one angle is 40°, then the opposite angle is 140°
Reason: Sum of opposite angles in a cyclic quadrilateral is equal to 360°
Answer: C

Question. Assertion: If length of a tangent from an external point to a circle is 8 cm, then length of the other tangent from the same point is 8 cm.
Reason: length of the tangents drawn from an external point to a circle are equal.
Answer: A

Fill in the Blanks

DIRECTIONS : Complete the following statements with an appropriate word / term to be filled in the blank space(s).

Question. A tangent to a circle touches it at ............... point (s).
Answer: One

Question. A line intersecting a circle at two points is called a ...........
Answer: Secant

Question. A circle can have .............. parallel tangents at the most.
Answer: Two

Question. The common point of a tangent to a circle and the circle is called .................. .
Answer: Point of contact

Question. There is no tangent to a circle passing through a point lying ............ the circle.
Answer: inside

Question. The tangent to a circle is .............. to the radius through the point of contact.
Answer: perpendicular

Question. There are exactly two tangents to a circle passing through a point lying ........... the circle.
Answer: outside

Question. The lengths of the two tangents from an external point to a circle are ............. .
Answer: equal

Question. The tangents drawn at the ends of a diameter of a circle are .................. .
Answer: Parallel

True / False

DIRECTIONS : Read the following statements and write your answer as true or false.

Question. The tangent to a circle is a special case of the secant.
Answer: True

Question. The perpendicular at the point of contact to the tangent to a circle does not pass through the centre.
Answer: False

Question. A circle can have at the most two parallel tangents.
Answer: True

Question. If P is a point on a circle with centre C, then the line drawn through P and perpendicular to CP is the tangent to the circle at the point P.
Answer: True

Question. The centre of the circle lies on the bisector of the angle between the two tangents.
Answer: True

Question. A tangent to a circle is a line that intersects the circle at only one point.
Answer: True

Question. Two equal chords of a circle are always parallel.
Answer: False

Question. A line drawn from the centre of a circle to a chord always bisects it.
Answer: False

Question. Line joining the centers of two intersecting circles always bisect their common chord.
Answer: True

Question. In a circle, two chords PQ and RS bisect each other. Then PRQS is a rectangle.
Answer: True

Write True or False and justify your answer in each of the following:

Question. Two chords AB and CD of a circle are each at distances 4 cm from the centre. Then AB = CD.
Answer: We know that chords equidistant from the centre of a circle are equal.
Here we are given that two chords AB and CD of a circle are each at distance 4 cm (equidistance) from the centre of a circle. So, chords are equal, i.e., AB = CD.
Hence, the given statement is true.

Question. Two congruent circles with centres O and O′ intersect at two points A and B. Then ∠AOB = ∠AO′ B.
Answer: The given statement is true because equal chords of congruent circles subtend equal angles at the respective centre. 

Question.  A circle of radius 3 cm can be drawn through two points A, B such that AB = 6 cm.
Answer: Radius of circle = 3 cm,
∴ Diameter of circle = 2 × r = 2 × 3 cm = 6 cm
Now, AB = 6 cm, so the given statement is true because AB will be the diameter.

Question.  ABCD is a cyclic quadrilateral such that ∠A = 90°, ∠B = 70°, ∠C = 95° and ∠D = 105°.
Answer: We know that opposite angles of a cyclic quadrilateral are supplementary.
Here, sum of opposite angles is not. 180º 
∠A+∠C = 90° + 95° =185°
Hence, ABCD is not a cyclic quadrilateral. The given statement is false.

Question. If A, B, C and D are four points such that ∠BAC = 45° and ∠BDC = 45°, then A, B, C, D are concyclic.
Answer: The given statement is true, because the two angles ∠BAC = 45° and ∠BDC = 45° are in the same segment of a circle.

Question. In Fig. 10.3, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to:
(A) 2 cm
(B) 3 cm
(C) 4 cm
(D) 5 cm

Answer: As perpendicular from the centre to a chord bisect the chord,

Question. In Fig.10.4, if ∠ABC = 20º, then ∠AOC is equal to:
(A) 20º
(B) 40º
(C) 60º
(D) 10º

Answer: Arc AC of a circle subtends AOC at the centre O and ABC at a point B on the remaining part of the circle,
∴ ∠AOC = 2∠ABC
= 2×20º ×40º
Hence, (b) is the correct answer.

Question. In Fig. 10.6, if ∠OAB = 40º, then ∠ACB is equal to:
(A) 50º
(B) 40º
(C) 60º
(D) 70°
Answer: In ΔOAB,
OA = OB [Radii of circle]
∴∠OAB = ∠OBA = 40º
[∵Angles opposite to equal sides are equal]

Question. ABCD is a cyclic quadrilateral such that AB is a diameter of the circle circumscribing it and ∠ADC = 140º, then ∠BAC is equal to:
(A) 80º
(B) 50º
(C) 40º
(D) 30º

Answer: ∠ADC +∠ABC =180º
 ⇒ 140º +∠ABC =180º
∠ABC =180º −140º = 40º
ABCD is a cyclic quadrilateral such that AB is the diameter of the circle circumscribing it.
Now, Join AC.∠C = 90º
[∵Angle in a semi-circle is a right angle]
InDABC,we have
∠BAC =180 (90º + 40º )
0 = 50º
Hence, (b) is the correct answer.

Question. In Fig. 10.8, BC is a diameter of the circle and ∠BAO = 60º. Then ∠ADC is equal to:
(A) 30º
(B) 45º
(C) 60º
(D) 120º

Answer: In ΔOAB, we have
OA = OB [Radii of the same circle]
∴ ∠ABO = ∠BAO [Angles opp. To equal sides are equal]
∴ ∠ABO = ∠BAO = 60º [Given]
Now, ∠ADC = ∠ABC = 60º
[∵ ∠ABC and ∠ADC are angles in the same segment of a circle, are equal]
Hence,  ∠ADC = 60º
So, (c) is the correct answer.

1. Two tangents PA and PB are drawn from an external point P to a circle with centre o. Prove that AOBP is a cyclic quadrilateral

2. Prove that the parallelogram circumscribing a circle is a rhombus

3. Two tangents PQ and PR are drawn to a circle with centre o from an external point P. `prove that Angle QPR = 2 angle OQR

4. If circle is inscribed in a ΔABC having sides 8 cm ,10 cm, 12 cm as shown in the figure. Find AD, BE and CF C

10cm F E 8 cm

A D B

12 cm

5. A circle is touching the side BC of a triangle ABC at P and AB and AC produced at Q and R respectively

Prove that AQ = AR = ½ perimeter of triangle ABC

6. in the isosceles ΔABC, AB = AC, show that BE = EC

E

B C

A

7. In the figure, PA and PB are tangents from P to the circle with centre O. LN touches the circle at M,

Then show that PL + LM = PN+ NM

A L

O M P

B N

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

9. Two concentric circles are of radii 7 cm and r cm, where r > 7. A chord of the larger circle, of length 48 cm touches the smaller circle. Find the value of r ( 25 cm)

10. In figure a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the tangents BD And DC into which BC is divided by the point of contact Dare the lengths 4 cm and 3 cm. If area of ΔABC = 21 cm2, then find the lengths of sides AB and AC (7.5 cm, 6.5 cm)

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

12. Two tangents PA and PB are drawn to the circle with centre o such that ˪APB = 120⁰. Prove that OP = 2 AP

13. Two concentric circles are of radii 13 cm and 5 cm. Find the length of the chord of the larger circle Which touches the smaller circle (24 cm)

14. Prove that the intercept of a tangent between a pair of parallel tangents to a circle subtend a right Angle at the centre of the circle

Please click the below link to access CBSE Class 10 Circles (2)