CBSE Class 10 Mathematics Circles VBQs Set 01

CIRCLE

1. There are 3 villages A, B and C such that the distance from A to B is 7 km, from B to C is 5 km and from C to A is 8 km. The gram pradhan wants to dig a well in such a way that the distance from each villages are equal. What should be the location of well? Which value is depicted by gram pradhan?

2. People of village wants to construct a road nearest to a circular village Rampur. The road cannot pass through the village. But the people wants that road should be at the shortest distance from the center of the village (i) which road will be the nearest to the center of village? (ii) which value is depicted by the people of village?

3. Four roads have to be constructed by touching village Khanpur in circular shape of radius 1700 m in the following manner.

Savita got contract to construct the roads AB and CD while Vijay got contract to construct AD and BC.

Prove that AB + CD = AD + BC. Which value is depicted by the contractor?

4. Two roads starting from P are touching a circular path at A and B.

Sarita ran from P to A 10 km and Ramesh ran from P to B. (i) If Sarita wins the race than how much distance Ramesh ran? (ii) Which value is depicted?

5. A farmer wants to divide a sugarcane of 7 ft length between his son and daughter equally. Divide it Geometrically, considering sugarcane as a line of 7cm, using construction.

(i) Find the length of each part.

(ii) Which value is depicted?
 

Question. From an external point Q, the length of the tangents to a circle is 5 cm and the distance of Q from the centre is 8 cm. The radius of the circle is:
(a) 39 cm
(b) 3 cm
(c) √39 cm
(d) 7 cm 
Answer : C

Question. From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is:
(a) 60 cm²
(b) 65 cm²
(c) 30 cm²
(d) 32.5 cm²
Answer : A

Question. The chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is ?
(a) 10 cm
(b) 10√2 cm
(c) 20 cm
(d) 12 cm
Answer : B

Question. At one end A of diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A is:
(a) 4 cm
(b) 5 cm
(c) 6 cm
(d) 8 cm
Answer : D

Question. If radii of concentric circles are 4 cm and 5 cm, then find the length of each chord of one circle which is tangent to the other circle.
(a) 8 cm
(b) 6 cm
(c) 10 cm
(d) 12 cm
Answer : B

Fill in the Blanks

Question. All concentric circles are ....................... to each other. 
Answer : Similar

Question. ABCD is a cyclic quadrilateral. If ∠BAC = 50° and ∠ÐDBC = 60° then ∠ BCD = ....................... .
Answer : 70º

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 = 13 cm. Then, the length PQ is ....................... .
Answer : 12 cm

Question. Secant intersects a circle at ....................... distinct points.
Answer : Secant intersects a circle at two distinct points.

Write True or False

Question. If a chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents at A and B is also 60°.
Answer : False.

Question. The length of tangent from an external point on a circle is always greater than radius of the circle.
Answer : False.

Question. The length of the tangent from an external point P on a circle with centre O is always less than OP.
Answer : True.

Question. The angle between two tangents to a circle may be 0°.
Answer : True.

Question. The tangent to the circum-circle of an isosceles ΔABC at A, in which AB = AC, is parallel to BC.
Answer : True.

Question. If a number of circles touch a given line segment PQ at a point A, then their centres lie on the perpendicular: bisector of PQ.
Answer : False.

Question. If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of PQ.
Answer : True.

Question. AB is the diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD.
Answer : True.
 

ASSERTION AND REASONING BASED QUESTION

Question : Assertion (A): PA and PB are two tangents to a circle with centre O. Such that ∠AOB = 1100, then ∠APB = 900.
Reason (R): The length of two tangents drawn from an external point are equal.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : D

Question : Assertion (A): 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 (R): (hypotenuse) 2 = (base) 2 + (height) 2
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : A

Question : Assertion (A): If a chord AB subtends an angle of 600 at the centre of a circle, then the angle between the tangents at A and B is also 600.
Reason (R): The length of the tangent from an external points P on a circle with centre O is always less than OP.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : D

Question : Assertion (A): The two tangents are drawn to a circle from an external point, then they subtend equal angles at the centre.
Reason (R):A parallelogram circumscribing a circle is a rhombus.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : B

Question : Assertion (A): The length of the tangent drawn from a point 8cm away from the centre of circle of radius 6cm is 2√7 cm.
Reason (R): If the angle between two radii of a circle is 1300, then the angle between the tangents at the end points of radii at their point of intersection is 500.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer : B