## AP 9th Class Maths Bits 9th Lesson Circles

Multiple Choice Questions (MCQs) :

Question 1.

In a circle with center O, the length of a chord AB is 16 and its distance from the center O is 6. What is the radius of the circle ?

A) 6

B) 8

C) 10

D) 12

Answer:

C) 10

Question 2.

In a circle, a chord of length 6 cm is at a distance of 4, cm from the center of the circle. What is the length of the radius of the circle ?

A) 2 cm

B) 3 cm

C) 4 cm

D) 5 cm

Answer:

D) 5 cm

Question 3.

In a circle, if a chord is of length 8 cm and the radius of the circle is 5 cm, what is the distance between the chord and the center of the circle?

A) 3 cm

B) 4 cm

C) 5 cm

D) 6 cm

Answer:

A) 3 cm

Question 4.

In a circle, if an angle at the center is 90 degrees, what is the measure of the subtended angle at the circumference?

A) 45 degrees

B) 90 degrees

C) 180 degrees

D) It cannot be determined from the given information.

Answer:

A) 45 degrees

Question 5.

In a circle, if the angle at the center is 60 degrees, what is the measure of the subtended angle at the circumference?

A) 30 degrees

B) 60 degrees

C) 120 degrees

D) 180 degrees

Answer:

A) 30 degrees

Question 6.

In a circle, the measure of an angle formed by a tangent and a chord is equal to

A) half the measure of the arc it subtends

B) the measure of the arc it subtends

C) twice the measure of the arc it subtends

D) the measure of the diameter of the circle

Answer:

A) half the measure of the arc it subtends

Question 7.

In a circle, the measure of an angle formed by two chords that intersect inside the circle is equal to

A) half the measure of the sum of the arcs they subtend

B) half the measure of the difference of the arcs they subtend

C) the measure of the sum of the arcs they subtend

D) the measure of the difference of the arcs they subtend

Answer:

B) half the measure of the difference of the arcs they subtend

Question 8.

In a circle, the measure of an angle formed by two chords that-intersect outside the circle is equal to

A) half the measure of the sum of the arcs they subtend

B) half the measure of the difference of the arcs they subtend

C) the measure of the sum of the arcs they subtend

D) the measure of the difference of the arcs they subtend

Answer:

B) half the measure of the difference of the arcs they subtend

Question 9.

In a circle with center O, if angle AOB is 60 degrees, what is the measure of angle ACB, where C is point on circumference of circle?

A) 90 degrees

B) 60 degrees

C) 30 degrees

D) 120 degrees

Answer:

C) 30 degrees

Question 10.

In a circle with center O, if angle AOB is 120 degrees, what is the measure of angle ACB, where C is point on circumference of circle?

A) 30 degrees

B) 60 degrees

C) 90 degrees

D) 120 degrees

Answer:

B) 60 degrees

Question 11.

In a circle with center O, if angle APB is 45 degrees, where P is point on circumference of circle. Then what is the measure of angle AOB, where AB is a chord of the circle?

A) 45 degrees

B) 90 degrees

C) 135 degrees

D) 180 degrees

Answer:

B) 90 degrees

Question 12.

In a circle with center O, if angle APB is 90 degrees, what is the measure of angle AOB, where AB is a chord of the circle?

A) 45 degrees

B) 90 degrees

C) 135 degrees

D) 180 degrees

Answer:

D) 180 degrees

Question 13.

In a circle with center O, if angle AOB is 120 degrees, what is the measure of angle APB, where P is point on circumference of circle, and AB is a chord of the circle ?

A) 45 degrees

B) 90 degrees

C) 135 degrees

D) 60 degrees

Answer:

B) 90 degrees

Question 14.

In a circle with center O, if angle APB is 66 degrees, what is the measure of angle AOB, where AB is a chord of the circle?

A) 60 degrees

B) 132 degrees

C) 150 degrees

D) 240 degrees

Answer:

B) 132 degrees

Question 15.

In a circle with center O, if angle AOB is 90 degrees, what is the measure of angle ACB, where AC and BC are tangents to the circle at points A and B, respectively?

A) 30 degrees

B) 45 degrees

C) 60 degrees

D) 90 degrees

Answer:

D) 90 degrees

Question 16.

In a circle with center O, if angle AOB is 60 degrees, what is the measure of angle ACB, where AC and BC are tangents to the circle at points A and B, respectively?

A) 120 degrees

B) 45 degrees

C) 60 degrees

D) 90 degrees

Answer:

A) 120 degrees

Question 17.

In a cyclic quadrilateral, opposite angles sure are ______________ .

A) supplementary

B) complementary

C) congruent

D) equal

Answer:

A) supplementary

Question 18.

If a quadrilateral is cyclic, then the sum of its opposite angles is ________ degrees.

A) 180

B) 360

C) 90

D) 270

Answer:

B) 360

Question 19.

The opposite angles of a cyclic quadrilateral are _______________ .

A) equal and complementary

B) equal and supplementary

C) congruent and complementary

D) congruent and supplementary

Answer:

D) congruent and supplementary

Question 20.

In a cyclic quadrilateral, the sum of two adjacent angles is ____________ degrees.

A) 90

B) 180

C) 270

D) 360

Answer:

B) 180

Question 21.

If the opposite angles of a quadrilateral are congruent, then the quadrilateral is _____________ .

A) a parallelogram

B) a rectangle

C) a square

D) a rhombus

Answer:

B) a rectangle

Question 22.

In a cyclic quadrilateral, the sum of any two opposite angles is _____________ degrees.

A) 180

B) 360

C) 90

D) 270

Answer:

A) 180

Question 23.

A quadrilateral is cyclic if and only if its opposite angles are _________________ .

A) congruent

B) supplementary

C) complementary

D) equal

Answer:

B) supplementary

Question 24.

In a cyclic quadrilateral, the measure of an angle subtended by an arc outside the quadrilateral is equal to the measure of the ____________ angle of the quadrilateral.

A) opposite

B) adjacent

C) diagonal

D) complementary

Answer:

A) opposite

Question 25.

What is the measure of the angle formed by a chord that is tangent to a circle at one of its endpoints?

A) 90 degrees

B) 180 degrees

C) 45 degrees

D) It depends on the length of the chord.

Answer:

A) 90 degrees

Question 26.

In a circle with a diameter of 10 cm, what is the measure of the angle subtended by the diameter?

A) 90 degrees

B) 180 degrees

C) 45 degrees

D) 120 degrees

Answer:

B) 180 degrees

Question 27.

In a circle with center O, what is the measure of the angle subtended by a semicircle at a point on circumference of circle?

A) 90 degrees

B) 180 degrees

C) 45 degrees

D) 120 degrees

Answer:

A) 90 degrees

Question 28.

If a chord of a circle subtends an angle of 60 degrees at the center of the circle, what is the measure of the angle subtended by the chord at point on circumference of circle ?

A) 30 degrees

B) 60 degrees

C) 90 degree

D) 120 degrees

Answer:

A) 30 degrees

Question 29.

If two chords of a circle intersect inside the circle, what is the relationship between the angles formed by the chord and their Intercepted arcs?

A) They are equal.

B) They are supplementary.

C) They are complementary.

D) They are congruent.

Answer:

B) They are supplementary.

Question 30.

In a circle, if two chord are equal in length, then

A) they are parallel

B) they are perpendicular

C) they are congruent

D) they intersect at the center of the circle

Answer:

C) they are congruent

Question 31.

The distance from the center of a circle to a chord is called

A) the radius

B) the diameter

C) the circumference

D) the chord length

Answer:

A) the radius

Question 32.

In a circle, if a chord is perpendicular to a radius, then

A) the chord bisects the radius

B) the chord is a diameter of the circle

C) the chord is tangent to the circle

D) the chord is a secant of the circle

Answer:

B) the chord is a diameter of the circle

Question 33.

In a circle, if two chords are equidistant from the center, then

A) they are parallel

B) they are perpendicular

C) they are congruent

D) they intersect at the center of the circle

Answer:

A) they are parallel

Question 34.

In a circle, an angle formed by two radii is called

A) a central angle

B) an inscribed angle

C) a tangent angle

D) a chord angle

Answer:

A) a central angle

Question 35.

In a circle, the measure of a central angle is equal to

A) the measure of the arc it subtends

B) half the measure of the arc it subtends

C) twice the measure of the arc it subtends

D) the measure of the diameter of the circle

Answer:

A) the measure of the arc it subtends

Question 36.

In a circle, the measure of an inscribed angle is equal to

A) half the measure of the arc it subtends

B) the measure of the arc it subtends

C) twice the measure of the arc it subtends

D) the measure of the diameter of the circle

Answer:

A) half the measure of the arc it subtends

Question 37.

The path traced by the tip of the second’s hand is a

A) circle

B) square

C) rectangle

D) straight line

Answer:

A) circle

Question 38.

The shape of the coin of ₹ 1 is

A) triangle

B) rhombus

C) circle

D) trapezium

Answer:

C) circle

Question 39.

The wheels of a vehicle are in

A) rectangular

B) triangular

C) circular

D) trapezoidal shape

Answer:

C) circular

Question 40.

The longest chord of a circle

A) Radius

B) Arc

C) Diameter

D) Segment

Answer:

C) circular

Question 41.

The centre of a circle lies

A) outside the circle

B) inside the circle

C) on the circle

D) none of these

Answer:

B) inside the circle

Question 42.

The minute hand of a clock is at 12 and the smaller hour’s hand is at 2. The angle between the hands of the clock is

A) 10°

B) 20°

C) 30°

D) 60°

Answer:

D) 60°

Question 43.

The perpendicular from the centre of a circle bisects the

A) circle

B) circumference

C) chord

D) radius

Answer:

C) chord

Question 44.

Given a circle with centre ‘O’ and smallest chord AB is of length 6 cm and the longest chord CD of the circle is of length 10 cm, then the radius of the circle is

A) 15 cm

B) 6 cm

C) 5 cm

D) 3.5 cm

Answer:

C) 5 cm

Question 45.

A chord of length 24 cm of a circle is at a distance of 5 cm from the centre. The radius of the circle is

A) 13 cm

B) 12 cm

C) 11 cm

D) 19 cm

Answer:

A) 13 cm

Question 46.

The length of a chord of a circle is 16 cm and its distance from the centre is 6 cm. The measure of radius of the circle is

A) 6 cm

B) 8 cm

C) 10 cm

D) 12 cm

Answer:

C) 10 cm

Question 47.

The length of the chord of a circle, of radius 13 cm, at a distance of 5 cm from the centre is

A) 12 cm

B) 18 cm

C) 20 cm

D) 24 cm

Answer:

D) 24 cm

Question 48.

In the following figure, ‘O is the centre of the circle. OA = 10 cm and perpendicular OC on chord AB = 8 cm, then the length of the chord AB is

A) 8 cm

B) 10 cm

C) 12 cm

D) 16 cm

Answer:

C) 12 cm

Question 49.

AD is a diameter of a circle and AB is a chord. If AD = 34 cm and 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:

D) 8 cm

Question 50.

In figure if OA = 5 cm, AB = 8 cm and OD ⊥ AB, then CD is equal to

A) 3 cm

B) 2 cm

C) 4 cm

D) 5 cm

Answer:

B) 2 cm

Question 51.

How many circles can pass through three given non-collinear points?

A) one and only one

B) two

C) three

D) infinitely many

Answer:

B) two

Question 52.

To determine a unique circle, the number of points required is

A) 1

B) 2

C) 3 non – coflinear points

D) 3 collinear points

Answer:

C) 3 non – coflinear points

Question 53.

How many points are sufficient to determine a line?

A) 1

B) 2

C) 3

D) none of these

Answer:

B) 2

Question 54.

Three chords AB, CD and EF of a circle are respectively 3 cm, 3.5 cm and 3.8 cm away tram the centre. Then which of the following is correct?

A) AB > CD > EF

B) AB < CD < EF

C) AB = CD = EF

D) AB = CD < EF

Answer:

A) AB > CD > EF

Question 55.

Equal chords of a circle are equidistant from

A) the centre

B) an extremity of a diameter

C) any point on the circumference

D) any point on the diameter

Answer:

A) the centre

Question 56.

The measure of the angle of a semi circle is

A) 30°

B) 45°

C) 60°

D) 90°

Answer:

D) 90°

Question 57.

The angle of a minor segment is

A) acute

B) right

C) obtuse

D) straight

Answer:

C) obtuse

Question 58.

The angle of a major segment is

A) acute

B) right

C) obtuse

D) straight

Answer:

A) acute

Question 59.

In the figure, AOB is a diameter of the semi circle, If ∠A s 60°, then ∠B is equal to

A)60°

B) 30°

C) 50°

D) 40°

Answer:

B) 30°

Question 60.

In the following figure, O is the centre of the circle PAB and ΔOAB is equilateral. The measure of ∠APB is equal to

A) 60°

B) 45°

C) 40°

D) 30°

Answer:

D) 30°

Question 61.

The length of a chord of a circle is equal to its radius. Find the measure of the angle subtended by that chord in major segment.

A) 30°

B) 60°

C) 45°

D) none of these

Answer:

A) 30°

Question 62.

∠ADB = 90° and ∠ABC = 30°, then ∠ACB is

A) 30°

B) 45°

C) 90°

D) 60°

Answer:

A) 30°

Question 63.

‘O’ is the centre of the circle, if ∠BOC – 140°, then x =

A) 70°

B) 40°

C) 35°

D) 20°

Answer:

D) 20°

Question 64.

The value of ‘x’ in figure is :

A) 35°

B) 45°

C) 55°

D) 30°

Answer:

A) 35°

Question 65.

In the following figure, there are two congruent circles whose centres are O and O’ and chord AB = chord CD. If ∠CO’D == 60°, then the measure of ∠APB is equal to

Answer:

A) 45°

B) 40°

C) 30°

D) 15°

Answer:

C) 30°

Question 66.

In the following figure, two circles C(O_{1}, r) and C(O_{2}, r) are congruent and arc \(\overparen{\mathbf{P Q}}\) = arc \(\overparen{\mathbf{R S}}\). If the measure of ∠RO_{2}S = 62° and the measure of ∠PAQ = x°, then the value of x is

A) 28°

B) 31°

C) 38°

D) 62°

Answer:

B) 31°

Question 67.

In the figure, ‘O’ is the centre of the circle. If ∠BAD = 48°, then ∠BCD =

Answer:

A) 96°

B) 48°

C) 42°

D) 84°

Answer:

B) 48°

Question 68.

In the figure, ‘O’ is the centre of the circle. The value of x is

A) 50°

B) 40°

C) 60°

D) 20°

Answer:

A) 50°

Question 69.

In the given figure, AD || BC and ∠BCA = 40°. The measure of ∠DBC is equal to

A) 50°

B) 80°

C) 40°

D) 20°

Answer:

C) 40°

Question 70.

In this figure AB = AC and ∠ABC = 50°. Then ∠BDC is equal to

A) 50°

B) 65°

C) 90°

D) 80°

Answer:

D) 80°

Question 71.

The opposite angles of a cyclic quadrilateral

A) are complementary

B) are supplementary

C) are equal

D) form a linear pair

Answer:

B) are supplementary

Question 72.

In the following figure, ABCD is a cyclic quadrilateral whose side AD is a diameter of the circle and the point O’ is the centre of the circle. If ∠OCD = 50°, then the measure of ∠ABC is

A) 100°

B) 120°

C) 110°

D) 130°

Answer:

D) 130°

Question 73.

In the figure, the magnitude of angle ABC if angle AOC = 120° will be

A) 125°

B) 120°

C) 130°

D) 135°

Answer:

B) 120°

Question 74.

In the below figure, ‘O’ is the centre of the circle and P, Q and R are points on , the circle such that ∠PQR = 100°, then ∠QPR equals :

A) 80°

B) 10°

C) 100°

D) 60°

Answer:

B) 10°

Question 75.

In the following figure, ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle. If ∠ADC = 120°, then the value of ∠CAB is

A) 30°

B) 45°

C) 60°

D) 90°

Answer:

A) 30°

Assertion and Reason type questions :

Question 1.

Assertion : If two circle in a circle are congruent, then the corresponding centred angles are also congruent.

Reason : The measure of an arc in a circle is equal to the measure of the central angle that subtends it.

Answer:

Assertion is true.

Reason is true and explains the assertion.

Explanation : The assertion is true because if two arcs in a circle are congruent, then they have the same length. Since the measure of an arc is equal to the length of the s arc divided by the radius of the circle, and the radius of the circle is the same for both arcs, it follows that the measures of the two arcs are equal. The reason is also true and explains why the corresponding central angles are congruent.

Question 2.

Assertion : If two angles are in the same segment of a circle, then they are congruent.

Reason : The angles in the same segment of a circle are subtended by the same chord.

Answer:

Assertion is true.

Reason is true and explains the assertion.

Explanation : The assertion is true because if two angles are in the same segment of a circle, then they are subtended by the same arc. Since the arc is part of a circle, it follows that the two angles are subtended by the same chord. Since the angles are inscribed in the same chord and subtend the same arc. they must be congruent. The reason is also true and explains why the angles in the same segment of a circle are congruent.

Question 3.

Assertion : In a cyclic quadrilateral, the opposite angles are supplementary.

Reason : The opposite angles in a cyclic quadrilateral are subtended by the same arc.

Answer:

Assertion is true.

Reason is true and explains the assertion :

Explanation : The assertion is true because a cyclic quadrilateral is a quadrilateral that

can be inscribed in a circle. This means that the opposite angles of the quadrilateral are inscribed in the same arc of the circle, and the measure of an inscribed angle is half the measure of the subtended arc. Since the opposite angles of a cyclic quadrilateral are inscribed in the same arc, they must have equal measures, and since their sum is equal to 180 degrees (the measure of the entire circle), it follows that they are supplementary. The reason is also true and explains why the opposite angles in a cyclic quadrilateral are supplementary.

Question 4.

Assertion : If a quadrilateral has two pairs of congruent adjacent sides, then it must be a cyclic quadrilateral.

Reason : A cyclic quadrilateral is a quadrilateral that can the inscribed in a circle.

Answer:

Assertion is false.

Reason is true.

Explanation : The assertion is false because a quadrilateral with two pairs of congruent adjacent sides is not necessarily a cyclic quadrilateral. For example, a rectangle has two pairs of congruent adjacent sides, but it is not a cyclic quadrilateral. The reason is true Because a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, and this definition does not depend on the lengths of its sides or angles.

Question 5.

Assertion : In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides.

Reason : The diagonals of a cyclic quadrilateral are perpendicular.

Answer:

Assertion is true.

Reason is false.

Explanation : The assertion is true because in a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides. This property is known as Ptolemy’s theorem. The reason is false because the diagonals of a cyclic quadrilateral are not necessarily perpendicular. However, it is true that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent, which is known as the circumcentre of the quadrilateral.

Fill in the blanks :

1. Two chords in a circle are congruent if ‘ and only if they are equidistant from the _____________ of the circle.

Answer:

Center

2. In a circle, the perpendicular bisector of a chord passes through the center of the circle if and only if the chord is a _______________ .

Answer:

Diameter

3. The measure of an angle formed by a chord and a tangent at one of its end points is equal to half the measure of the _____________ .

Answer:

Intercepted arc

4. Two chords intersect inside a circle. If the products of their segments are equal, then the chords are ______________ .

Answer:

Equidistant

5. If two chords in a circle are parallel, then the segments they cut off on the circle are _______________ .

Answer:

Congruent

6. The angle formed by two intersecting chords in a circle is equal to the average of the measures of the arcs ____________ and _______________ .

Answer:

Intercepted by the chords section

7. If two chords in a circle are perpendicular, then the product of the segments of one chord is equal to the product of the segments of the ________________ .

Answer:

Other chord

8. The measure of an angle formed by two intersecting chords in a circle is equal to half the sum of the measures of the arcs ______________ and _____________ .

Answer:

Intercepted by the chord section

9. A quadrilateral is cyclic if and only if its opposite angles are ____________ .

Answer:

Equal

10. In a cyclic quadrilateral, the sum of the measures of opposite angles is ______________ degrees.

Answer:

180

11. If a quadrilateral is cyclic, then its opposite angles are ______________ .

Answer:

Supplementary

12. A quadrilateral is cyclic if and only if its interior opposite angles are _______________ .

Answer:

Supplementary

13. In a cyclic quadrilateral, the sum of any two adjacent angles is ______________ degrees.

Answer:

180

14. The opposite angles of a cyclic quadrilateral are ______________ .

Answer:

Congruent

15. In a cyclic quadrilateral, the sum of any two opposite angles is ____________ degrees.

Answer:

180

16. A quadrilateral is cyclic if and only if a pair of opposite angles are ____________ .

Answer:

Supplementary

17. The measure of an angle subtended by an arc outside the quadrilateral is equal to the measure of the ____________ angle of the quadrilateral.

Answer:

Inscribed

18. In a cyclic quadrilateral, if one angle is right, then the opposite angle is ______________ .

Answer:

Right

19. Angles in the same segment of a circle ______________ .

Answer:

Equal

20. In a cyclic quadrilateral, the sum of pair of opposite angles is ______________ .

Answer:

180°