CBSE Class 10 Mathematics Circles Worksheet Set B

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.

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.

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

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