CBSE Class 10 Mathematics Circles Worksheet Set A

Question. AD is a diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, the distance of AB from the centre of the circle is:
(A) 17 cm
(B) 15 cm
(C) 4 cm
(D) 8 cm
 

Answer : Draw OP ⊥ AB.
As perpendicular from the centre to a chord bisect the chord, so

Question. If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is:
(A) 6 cm
(B) 8 cm
(C) 10 cm
(D) 12 cm

Answer : AB is perpendicular to BC, therefore ABC is a right triangle.

In right ΔABC , we have

Question. In Fig.10.5, if AOB is a diameter of the circle and AC = BC, then ∠CAB is equal to:
(A) 30º
(B) 60º
(C) 90º
(D) 45º

Answer : As AOB is a diameter of the circle,
∠C = 90º
[∵Angles in a semi-circle is 90º]
Now, AC = BC
∠A = ∠B
[∵Angles opposite to equal sides of triangle are equal]
0 ∠A+∠B +∠C =180º
 ⇒2∠A+ 90º =180º
⇒ 2∠A = 90º⇒∠A = 90º ÷ 2 = 45º
Hence, (d) is the correct answer.

Question. In Fig. 10.7, if ∠DAB = 60º, ∠ABD = 50º, then ∠ACB is equal to:
(A) 60º
(B) 50º
(C) 70º
(D) 80º

Answer : In ΔADB, we have
∠A+∠B +∠D =180
⇒ 60º + 50º +∠D =180º
⇒ ∠D =180 −110 = 70º
i.e., 0 ∠ABD = 70º
Now, ∠ACB = ∠ADB = 70º
[∵Angles in the same segment of a circle are equal]
Hence, (c) is the correct answer.

10. In Fig. 10.9, ∠AOB = 90º and ∠ABC = 30º, then ∠CAO is equal to:
(A) 30º
(B) 45º
(C) 90º
(D) 60º

Answer : In ΔOAB, we have
OA = OB
[Radii of the same circle]
∴∠OAB = ∠OBA
∴ 2∠OAB = (180º −∠AOB)
= (180º −90º ) [∵Sum of angles of Δ is 180º]

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

Question. Two chords AB and AC of a circle with centre O are on the opposite sides of OA.
Then ∠OAB = ∠OAC.
Answer : The given statement is false, because the angles will be equal if AB = AC.

Question. Through three collinear points a circle can be drawn.
Answer : The given statement is false because a circle through two points cannot pass through a point which is collinear to these two points.

Question. If AOB is a diameter of a circle and C is a point on the circle, then AC² + BC² = AB²
Answer : AOB is a diameter of a circle and C is a point on the circle.
∴ ∠ACB = 90º [∵Angle in a semicircle is a right angle]
In right ΔABC,
AC² + BC² = AB² [By Pythagoras theorem]
Hence, the given statement is true.

Question. If A, B, C, D are four points such that ∠BAC = 30° and ∠BDC = 60°, then D is the centre of the circle through A, B and C.
Answer : The given statement is false because there can be many points D such that ∠BDC = 60° and each such point cannot be centre of the circle through A, B, C.

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1.At how many point does a tangent intersect to a circle?

(A) One

(B) Two

(C) Three

(D) Infinite

2.From a point P a tangent is drawn to circle of diameter 48 cm. The point P is situated at a distance of

25 cm from center O of the circle then the length of tangent is:

(A) 7 cm

(B) 14 cm

(C) 16 cm

(D) 24 cm

3.Two tangents are drawn at the end of a diameter of a circle.What is the distance between diameter if

the area of circle is 154 cm2 ?

(A) 7 cm

(B) 14 cm

(C) 21 cm

(D) 28 cm

4.From a point Q the length of the tangent to a circle is 24 cm and radius of circle is 7 cm then the

distance of a Q from center is:

(A) 12 cm

(B) 12.5cm

(C) 25 cm

(D) 50 cm

5.If two tangents from point P are drawn to circle at points Q and R, if they are inclined at 100° then

QOR equal to (where O is center of circle)

(A) 70°

(B) 80°

(C) 90°

(D) 100°

6.From a point Q the length of tangent to circle is 24 cm and distance Q from the center is 25 cm then

the area of circle is:

(A) 7

(B) 14

(C) 49

(D) None of these

7.Two centric circles are of radii 25 cm and 24 cm. then what is the length of the chord of the larger

circle which touches the smaller circle?

(A) 7 cm

(B) 14 cm

(C) 21 cm

(D) 28 cm

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