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All questions of Circles for Class 9 Exam

In the figure, AD is a straight line. OP is perpendicular to AD and O is the centre of both the circles. If AO = 34 cm, OB = 20 cm and OP = 16 cm, then the length of AD is
a)
20 cm
b)
50 cm
c)
60 cm
d)
43 cm
Correct answer is option 'C'. Can you explain this answer?

Sarthak Satav answered

Here OPA is right angle triangle given OP = 16 cm AO = 34 cm AP = ? using Pythagoras theorem : AO² = OP² + AP² 34² = 16² + AP² 1156 = 256 + AP² 1156 - 256 = AP² 900 = AP² *√900 = AP* 30 = AP Since AD is a straight line and O is at center of both circles AD = AP + PD AD = AP + AP AD = 2AP AD = 2 x 30 *AD = 60 CM*

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The given figure shows two intersecting circles. If ∠ABC = 75^o, then the measure of ∠PAD is
a)
75^o
b)
125^o
c)
105^o
d)
150^o
Correct answer is option 'C'. Can you explain this answer?

Zachary Foster answered

In the given figure, the points A, B, C, and D form a cyclic quadrilateral because they lie on the circumferences of the two intersecting circles.

The opposite angles of a cyclic quadrilateral are supplementary, i.e.,

It is given that ∠ABC=75^∘

Using the property of cyclic quadrilaterals:

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P is a point on the diameter AB of a circle and CD is a chord perpendicular to AB. If AP = 4 cm and PB = 16 cm, the length of chord CD is

a)
16 cm

b)
20 cm

c)
10 cm

d)
8 cm

Correct answer is option 'A'. Can you explain this answer?

EduRev Class 9 answered

AP+PB=AB [diameter of circle]
4+16=20[diameter]
so r=AO = CO=10
so clearly PO= OA-AP=10-4=6
PO=6 OC=10 So just apply pythagoreus theorem in triangle PCO
so CP will come 8 So chord CD = 2x8 = 16

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If a chord of a circle is equal to its radius, then the angle subtended by this chord in major segment is
a)
45^o
b)
30^o
c)
60^o
d)
90^o
Correct answer is option 'B'. Can you explain this answer?

Zachary Foster answered

Let AB be the chord of the circle with center O

Given that AB = Radius of the circle.

Also, AO = BO = Radius

∴ ΔOAB is an equilateral triangle.

Thus, ∠AOB = ∠OBA = ∠OAB = 60°

Also, angle subtended by an arc at the center of the circle is twice the angle subtended by it at any other point in the remaining part of the circle.

∴ ∠AOB = 2∠ACB

⇒ ∠ACB = 1/2 (∠AOB)

⇒ ∠ACB = 1/2 (60°) = 30°

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In the given figure, AD is the diameter of the circle and AE = DE. If ∠ABC = 115^o, then the measure of ∠CAE is

a)
70^o
b)
90^o
c)
80^o
d)
60^o
Correct answer is option 'A'. Can you explain this answer?

Nclex Coaching Centre answered

Since AD is the diameter, the angle subtended by the diameter at any point on the circle is always 90^∘. Therefore:

∠ACD = 90^∘

Given AE = DE, triangle ADE is isosceles. Therefore:

∠DAE =∠DEA

Since ABCD is a cyclic quadrilateral, opposite angles of a cyclic quadrilateral sum to 180^∘:

∠ABC + ∠ADC = 180^∘

Substituting ∠ABC=115^∘

∠ADC =180^∘−115^∘= 65^∘

In triangle ADE, since AE = DE, we have:

∠DAE=∠DEA

As ∠ADC=65^∘(external angle of triangle ADE), we know that:

∠DAE = 1 / 2 × 65^∘= 32.5^∘

Finding ∠CAE: Since ∠CAE=∠DAE, we find:

∠CAE = 70^∘

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Chords AB and CD intersect at right angles. If ∠BAC = 40^o, then ∠ABD is equal to
a)
45^o
b)
50^o
c)
60^o
d)
none of these
Correct answer is option 'B'. Can you explain this answer?

Zachary Foster answered

The correct answer is 50°

GIVEN: Chords AB and CD intersect at right angle. ∠BAC = 40°

TO FIND: ∠ABD

SOLUTION

We can simply solve the above problem as follows;

Let the point of intersection of the chord be 'O'.

∠AOC = 90°

Therefore,

In ΔAOC

∠AOC + ∠OAC + ∠ACO = 180°

90 + 40 + ∠ACO = 180

∠ACO = 180- 130 = 50°

Now,

AO :CO = 5:4 [since larger angle denotes shorter length]

Therefore,

Ratio of-

OB:OD=4:5 [ since they intersect and ratio gets changed]

or,

∠OBD:∠ODB=5:4 [larger length has smaller ratio]

Therefore,

∠OBD+∠ODB+90=180

5x + 4x + 90 = 180

9x + 90 = 180

x + 10 = 20

x = 10

∠OBD = 5 × 10 = 50°

Since,

∠OBD = ∠ABD

Therefore,

∠ABD = 50°

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A circle with centre O in which diameter AB bisects the chord CD at point E. If CE = ED = 8 cm and EB = 4 cm, the radius of the circle is
a)
10 cm
b)
8.5 cm
c)
16 cm
d)
20 cm
Correct answer is option 'A'. Can you explain this answer?

Maroof Khan answered

Option A

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In the given figure, O is the centre of the circle ABE is a straight line,. If ∠DBE = 95^o then ∠AOD is equal to
a)
190^o
b)
170^o
c)
180^o
d)
175^o
Correct answer is option 'B'. Can you explain this answer?

Zachary Foster answered

AE is a straight line so,

angle ABD + angle EBD = 180°

angle ABD + 95° = 180°

angle ABD = 180° - 95°

angle ABD = 85°

so, it is interior angle of circle

So, angle AOD is double of angle suspended on the circle

angle AOD = 2ABD

angle AOD = 2(85°)

angle AOD = 170°

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XY and PQ are two parallel chords of a circle on opposite side of the centre O and radius of circle is 13 cm. if PQ = 10 cm and XY = 24 cm, find the distance between PQ and XY.
a)
17 cm
b)
14 cm
c)
25 cm
d)
21 cm
Correct answer is option 'A'. Can you explain this answer?

Cat Warrior.ÀloŇƏ answered

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X is a point on a circle with centre O. If X is equidistant from the two radii OP, OQ, then arc PX : arc PQ is equal to
a)
it is 1 : 2
b)
it is 1 : 1
c)
it is 2 : 1
d)
it is 2 : 3
Correct answer is option 'A'. Can you explain this answer?

Skyhigh Academy answered

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Chords AD and BC intersect each other at right angles at point P. ∠DAB = 35^o, then ∠ADC is equal to
a)
55^o
b)
45^o
c)
35^o
d)
65^o
Correct answer is option 'A'. Can you explain this answer?

Madhu answered

Option a

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In the given figure ∠SOR = 37.5° find the value of ∠PTQ.

a)
60°

b)
67°

c)
45°

d)
71.25°

Correct answer is option 'D'. Can you explain this answer?

EduRev Class 9 answered

∠SOR = 37.5°

∠SQT = 1/2 ∠SOR (Angle at the circumference is half of the angle at the centre)

⇒ 37.5°/2

⇒ 18.75°

∠QSP = 90° (angle made from the diameter to the circumference is 90°)

Now,

∠PSQ + ∠QST = 180° (Linear pair angle)

∠QST = 180° - 90° = 90°

In ∆QST,

∠STQ = 180° - (90° + 18.75°)

⇒ ∠STQ = 71.25°

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In the given figure, chords AB and CD intersect each other at right angles. Then, ∠x+∠y is equal to
a)
75^o
b)
90^o
c)
45^o
d)
60^o
Correct answer is option 'B'. Can you explain this answer?

Zachary Foster answered

In the circle, AB and CD are two chords which intersect each other at P at right angle i.e. ∠CPB=90^o.

∠CAB and ∠CDB are in the same segment.

∴ ∠CDB=∠CAB=x

Now, in ΔPDB, Ext. ∠CPB=∠D+∠DBP

⇒90^o= x+y (∵CD⊥AB)

Hence, x+y = 90^o

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In the given figure PQ = QR = RS and ∠PTS = 75^o then the measure of ∠QOR is
a)
25^o
b)
20^o
c)
50^o
d)
75^o
Correct answer is option 'C'. Can you explain this answer?

Lalit Jain Jain answered

D) 75

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Two equal chords AB and CD of a circle are such that the length of perpendicular OE on CD = 5 cm. If OF is the perpendicular on AB, then OF =
a)
5 cm
b)
2.5 cm
c)
10 cm
d)
15 cm
Correct answer is option 'A'. Can you explain this answer?

Rudra Shandilya answered

Equal chords are equidistant from the centre, so length of OF=OE=5 cm

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In the given figure, ABCD is a cyclic quadrilateral in which ∠BAD = 75^o, ∠ABD = 58^o and ∠ADC = 77^o , AC and BD intersect at P. the measure of ∠DPC is

a)
92^o

b)
105^o

c)
90^o

d)
94^o

Correct answer is option 'A'. Can you explain this answer?

Varsha Pawar answered

ANS:- let ∠PBC = x

∠ADC + ∠ABC = 180... ( opposite angles of a cyclic quadrilateral are supplementary)

77+ ∠ABD + x =180.. (∠ABC =∠ABD + x)

77+58+x =180.
135 + x =180
x = 180-135
x =45
∠PBC=45..(1)

Now , in ∆ADB,

∠ABD+∠BAD+ ∠ADB=180

58+75+ ∠ADB=180.

133+∠ADB=180.

∠ADB=180-133

∠ADB=47.

Here,. ∠ACB = ∠ADB... (angles in the same segment are equal)

Therefore,. ∠ACB=47....(2)

Now in ∆DPC,
∠DPC = ∠ACB + ∠PBC ..(exterior angle theoram.)

∠DPC= 47+45.... [from(1) and (2)]

Therefore,. ∠DPC=92.

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BC is a diameter of the circle and ∠BAO = 60^o . Then ∠ADC is equal to
a)
45^o
b)
30^o
c)
60^o
d)
90^o
Correct answer is option 'C'. Can you explain this answer?

Swati Malkani answered

Option C is correct i.e 60 degree

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In the given figure, AC is a diameter of the given circle and ∠BCD = 75^o. Then, ∠EAF−∠ABC is equal to
a)
15^o
b)
25^o
c)
20^o
d)
10^o
Correct answer is option 'A'. Can you explain this answer?

Learning Education answered

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The region between chord and either of the arc is called
a)
a sector
b)
a semicircle
c)
a segment
d)
a quarter circle
Correct answer is option 'C'. Can you explain this answer?

Nclex Coaching Centre answered

The region between a chord and either of its arcs is called a segment of the circular region or simply a segment of the circle. The segment formed by minor arc along with chord, is called minor segment and the segment formed by major arc, is called the major segment.

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In the given figure, O is the centre of the circle. ∠OAB and ∠OCB are 40^o and 30^o respectively. Then, the measure of ∠AOC is
a)
170^o
b)
120^o
c)
110^o
d)
140^o
Correct answer is option 'D'. Can you explain this answer?

C K Academy answered

(Angle at the centre is double the angle at the circumference subtended by the same chord)

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In the given figure if ∠CAB = 49^o and ∠ADC = 43^o, then the measure of ∠ACB is
a)
92^o
b)
88o
c)
74^o
d)
96^o
Correct answer is option 'B'. Can you explain this answer?

Nawaz Khan answered

X+ 43+ 49 = 180
x+92= 180
x= 88

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ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. If ∠BEF = 80^o, then ∠ABC is equal to
a)
120^o
b)
100^o
c)
80^o
d)
75^o
Correct answer is option 'C'. Can you explain this answer?

Let's Tute answered

Given, ABCD is a parallelogram and AEFD is a cyclic quadrilateral.
∠BEF=80^∘
Now, ∠ADC=∠BEF=80^∘ (Angle of a cyclic quadrilateral is equal to the opposite exterior angle )
Also, now in parallelogram ABCD,
∠ABC=∠ADC=80^∘ (Opposite angles of a parallelogram are equal)

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In the given figure, O is the centre of the circle. If ∠QPR is 50^o, then ∠QOR is :
a)
130^o
b)
50^o
c)
100^o
d)
40^o
Correct answer is option 'C'. Can you explain this answer?

Swati Malkani answered

Option C is correct

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In the figure, angle ABC = 65° and angle ACB = 58°, the measure of angle BDC is
a)
45°
b)
68°
c)
90°
d)
57°
Correct answer is option 'D'. Can you explain this answer?

Nclex Coaching Centre answered

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Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. The length of the common chord is
a)
5 cm
b)
8 cm
c)
6 cm
d)
10 cm
Correct answer is option 'C'. Can you explain this answer?

Zachary Foster answered

Let the radius of the circle centered at O and O' be 5 cm and 3 cm respectively.

OA = OB = 5 cm

O'A = O'B = 3 cm

OO' will be the perpendicular bisector of chord AB.

∴ AC = CB

⇒ 5²= AC² + x²

⇒ 25 − x² = AC² ...(1)

In ΔO'AC,

O'A2 = AC2 + O'C2

⇒ 3² = AC² + (4 − x)²

⇒ 9 = AC² + 16 + x² − 8x

⇒ AC² = − x²− 7 + 8x ... (2)

From equations (1) and (2), we obtain

25 − x² = − x² − 7 + 8x

8x = 32

x = 4

Therefore, the common chord will pass through the centre of the smaller circle i.e., O' and hence, it will be the diameter of the smaller circle.

AC² = 25 − x² = 25 − 42 = 25 − 16 = 9

∴ AC = 3 m

Length of the common chord AB = 2 AC = (2 × 3) m = 6 m

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Angle formed in minor segment of a circle is
a)
a straight angle
b)
an acute angle
c)
a right angle
d)
an obtuse angle
Correct answer is option 'D'. Can you explain this answer?

Zachary Foster answered

Whenever a chord is drawn in a circle two segments are formed. One is called the minor segment while the other is called the major segment.
The angle formed by the chord in the minor segment will always be obtuse.

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The perpendicular distance of a chord 8 cm long from the centre of a circle of radius 5 cm is
a)
2 cm
b)
9 cm
c)
4 cm
d)
3 cm
Correct answer is option 'D'. Can you explain this answer?

Nclex Coaching Centre answered

Consider a circle having center O with a chord. Let OA be the radius of the circle and AB be the chord. As given in the question, the radius of the circle is 5 cm and length of chord is 8 cm. Let the distance between the center of the circle and chord be OP. So, this can be shown diagrammatically as:

It is clear from the diagram that OP is perpendicular to AB. As OP is perpendicular to AB and passes through the center O, it will bisect the chord AB at P. Now the length of AP will be,
AP =1/2 × AB

AP = 1/2 × 8

AP = 4 cm
Since, triangle OPA is a right-angle triangle, we can easily apply the Pythagoras theorem which can be stated as b²+p²= h² where b, p and h are base, perpendicular and hypotenuse of the respective triangle.
In ΔOPA,

AP²+OP²=AO²

OP²= AO²− AP²

OP²= 5²−4²

OP = 3 cm
Therefore, the distance of the chord AB from the center is 3 cm.

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In the given figure, angle OAC is 35°, angle ADC is
a)
45°
b)
60°
c)
55°
d)
50°
Correct answer is option 'C'. Can you explain this answer?

Nclex Coaching Centre answered

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The radius of a circle which has a 6 cm long chord, 4 cm away from the centre of the circle is
a)
9 cm
b)
8 cm
c)
10 cm
d)
5 cm
Correct answer is option 'D'. Can you explain this answer?

Zachary Foster answered

In the right triangle OAP,

OA² = OP² +AP² (By Pythagoras theorem)

OA² = 4² + 3² (perpendicular from the centre of the circle bisects the chord , AP=3cm)

OA² = 25

OA = 5

Hence the radius of the circle is 5 cm.

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In the given figure, a circle is centred at O. The value of x is :
a)
55^o
b)
70^o
c)
110^o
d)
125^o
Correct answer is option 'C'. Can you explain this answer?

Nclex Coaching Centre answered

Use the Property for Angles Subtended at the Center: The total angle subtended by the arc AB at the center O is given by:

∠AOB = 2 × ∠CAB + 2 × ∠CBA

Substitute the Given Values:

∠AOB = 2 × 20^∘ + 2 × 35^∘

Simplify: ∠AOB = 40^∘+ 70^∘= 110^∘

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In the given figure, AB is a diameter of the circle APBR. APQ and RBQ are straight lines. If ∠A = 35^o and , then the measure of ∠PBR is
a)
165^o
b)
115^o
c)
155^o
d)
135^o
Correct answer is option 'B'. Can you explain this answer?

Nclex Coaching Centre answered

Angle BPA = 90°

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In the given circle, O is the centre and ∠BDC = 42^o. Then, ∠ACB is equal to
a)
58^o
b)
42^o
c)
52^o
d)
48^o
Correct answer is option 'D'. Can you explain this answer?

Pooja Shah answered

In ∆ BDC and ∆ BAC
Angle BAC = BDC
(angle made on same segment BC)
Since ABC is making right angle (90)
So,
In ∆ABC
ABC +BAC+ACB=180
(angle sum property of triangle)
90+42+ACB=180
ACB=180-132
ACB=48^o

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In the given, AB is side of regular five sided polygon and AC is a side of a regular six sided polygon inscribed in the circle with centre O. AO and CB intersect at P, then ∠APB is equal to
a)
72^o
b)
98^o
c)
90^o
d)
96^o
Correct answer is option 'D'. Can you explain this answer?

Nclex Coaching Centre answered

For a regular n-sided polygon inscribed in a circle, the central angle subtended by each side is:

Central Angle = 360^∘ / n

For the pentagon (n=5), the central angle subtended by AB is:

360^∘ / 5 =72^∘

For the hexagon (n=6), the central angle subtended by AC is:

360^∘/ 6 = 60^∘

The angle ∠APB between the two intersecting chords AB and AC can be calculated using the property of angles formed by intersecting chords: ∠APB = 1 /2 × (Sum of the arcs subtended by the opposite segments)

Arc subtended by AB: 72^∘
Arc subtended by AC: 60^∘
The sum of the opposite arcs involved is: 72^∘+120^∘=192^∘

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AOB is the diameter of the circle. If ∠AOE = 150^o, then the measure of ∠CBE is
a)
120^o
b)
125^o
c)
105^o
d)
115^o
Correct answer is option 'C'. Can you explain this answer?

Swati Malkani answered

Option C is correct.

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The length of chord which is at a distance of 12 cm from centre of circle of radius 13 cm is:
a)
12 cm
b)
13 cm
c)
5 cm
d)
10 cm
Correct answer is option 'D'. Can you explain this answer?

Rajput Sanjanajamwal San answered

The drawn right triangle;

Hypotenuse H= Radius of circle =13 cm.

P= perpendicular distance =12 cm, and

′L′ be the length of half-chord

(Note that By the property of Circle, Perpendicular drawn from Centre to Chord Bisects the Chord.)

By Pythagoras theorem, in the drawn right triangle shown in figure,

H2=L2+P2

L=132–122=5 cm

Now, length of the chord is 2.L=10cm

Therefore the length of chord will be 10 cm.

So, option D will be the answer.

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In the given figure, the measure of angle BCD is
a)
100°
b)
75°
c)
60°
d)
50°
Correct answer is option 'C'. Can you explain this answer?

Nclex Coaching Centre answered

The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circumference.

∠ABC=70^∘, subtended by the arc AC.
Therefore, the angle subtended by the same arc AC at the circumference on the opposite side (∠BCD) can be calculated.

∠BCD = 180^∘ − (∠BAD + ∠ABC)

∠BCD = 180^∘ − (50^∘+70^∘)

∠BCD = 60^∘

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Number of circles that can be drawn through three non-collinear points is
a)
3
b)
1
c)
2
d)
0
Correct answer is option 'B'. Can you explain this answer?

Zachary Foster answered

1. Understanding Non-Collinear Points:
- Non-collinear points are points that do not all lie on the same straight line. For example, if we have three points A, B, and C, they form a triangle if they are non-collinear.

2. Circle through Two Points:
- If we take any two points, say A and B, an infinite number of circles can be drawn through these two points. This is because circles can be drawn with different radii and centers that still pass through points A and B.

3. Adding the Third Point:
- When we add a third point C, which is not on the line formed by A and B, we can only draw one unique circle that passes through all three points A, B, and C. This is because a circle is uniquely defined by three non-collinear points.

4. Conclusion:
- Therefore, the number of circles that can be drawn through three non-collinear points is exactly one.

Final Answer: The answer is (a) 1.

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as

Assertion : The measure of ∠AOC = 60º

Reason : Angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by arc on the circumference.

a)
Both assertion and reason are true and reason is the correct explanation of assertion.

b)
Both assertion and reason are true but reason is not the correct explanation of assertion.

c)
Assertion is true but reason is false.

d)
Assertion is false but reason is true.

Correct answer is option 'D'. Can you explain this answer?

Meera Rana answered

Join BO.

In ΔAOB , we have

OA = OB [radius]

∠OBA = ∠OAB

[Angle opposite to equal sides of a triangle are equal]

∠OBA = 30º ...(1)

Similarly, in ΔBOC , we get OB = OC

∠OCB = ∠OBC

∠OBC = 40º ...(2)

∠ABC = ∠OBA + ∠OBC

= 30º + 40º = 70º

[Using (1) and (2)]

Since angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by the arc on the circumference.

∠AOC = 2 x ∠ABC

= 2 x 70º = 140º
As the Assertion states that ∠AOC=60^∘, it is false.

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : Two diameters of a circle intersect each other at right angles. Then the quadrilateral formed by joining their end-points is a square.
Reason : Equal chords subtend equal angles at the centre.
a)
Both assertion and reason are true and reason is the correct explanation of assertion.
b)
Both assertion and reason are true but reason is not the correct explanation of assertion.
c)
Assertion is true but reason is false.
d)
Assertion is false but reason is true
Correct answer is option 'A'. Can you explain this answer?

Raina Kapoor answered

Assertion: Two diameters of a circle intersect each other at right angles. Then the quadrilateral formed by joining their end-points is a square.

Reason: Equal chords subtend equal angles at the centre.

Explanation:
To understand the given assertion and reason, let's break down the information provided and analyze it step by step.

1. Two diameters of a circle intersect each other at right angles:
- A diameter is a line segment passing through the center of a circle and having both its endpoints on the circle.
- When two diameters intersect each other, they form four angles at their intersection point, also known as the center of the circle.
- Since the diameters intersect at right angles, the four angles formed are all right angles.

2. The quadrilateral formed by joining their end-points is a square:
- A quadrilateral is a polygon with four sides.
- If the four angles of a quadrilateral are all right angles, then it is a square.
- In this case, the four angles formed at the intersection of the diameters are all right angles, so the quadrilateral formed is a square.

3. Equal chords subtend equal angles at the center:
- A chord is a line segment that connects two points on the circumference of a circle.
- If two chords of a circle are equal in length, then the angles they subtend at the center of the circle are also equal.
- This property is useful in determining the congruence of angles formed by chords in a circle.

Now, let's analyze the relationship between the assertion and the reason:

- The assertion states that when two diameters of a circle intersect at right angles, the quadrilateral formed is a square.
- The reason states that equal chords subtend equal angles at the center.
- In this case, the two diameters of the circle intersect at right angles, forming equal chords.
- Since the equal chords subtend equal angles at the center, the four angles formed at the intersection of the diameters are all equal and right angles.
- Therefore, the reason correctly explains why the quadrilateral formed is a square.

Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion. Therefore, the correct answer is option A.

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : The length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm is 17.32 cm.
Reason : The perpendicular from the centre of a circle to a chord bisects the chord.
a)
Both assertion and reason are true and reason is the correct explanation of assertion.
b)
Both assertion and reason are true but reason is not the correct explanation of assertion.
c)
Assertion is true but reason is false.
d)
Assertion is false but reason is true.
Correct answer is option 'A'. Can you explain this answer?

Aditi Sharma answered

Let, PQ be a chord of a circle with centre O and radius 10cm. Draw OR ⊥PQ = .

Now, OP = 10cm and OR = 5cm

In right triangle ORP, we get

OP² = PR² + OR²

PR2 = OP² - OR²

PR² = 10² - 5² = 75

Since, the perpendicular from the centre to a chord bisects the chord.

Therefore, PQ = 2 x PR

= 2 x 8.66 = 17.32 cm

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AD is a diameter of a circle and AB is chord. If AB = 24 cm, AD = 30 cm, the distance of AB from the centre of the circle is :

a)
9 cm

b)
8 cm

c)
17 cm

d)
15 cm,

Correct answer is option 'A'. Can you explain this answer?

Sarthak Satav answered

AD = 30 cm AO = 30 ÷ 2 CM AO = 15 CM since m is perpendicular (90 degree and joins O ) AM = AB ÷ 2 AM = 24 ÷ 2 AM = 12 CM Using Pythagoras theorem AO² = OM² + AM² 15² = OM² + 12² 225 - 144 = OM² 81 = OM² √81 = OM OM = 9 CM Hence "A" is right answer

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Two circles of radii10 cm and 8 cm intersect and the length of the common chord is 12 cm. The distance between their centers is

a)
3 cm

b)
6 cm

c)
(8+2√7) cm

d)
5 cm

Correct answer is option 'C'. Can you explain this answer?

Rohan Rane answered

Given length of common chord AB =12 cm

Let the radius of the circle with centre O is OA = 10 cm

Radius of circle with centre P is AP = 8 cm

From the figure, OP⊥AB

⇒AC = CB

∴AC = 6 cm (Since AB=12 cm)

In ΔACP,

AP²= PC²+AC² [By Pythagoras theorem]

⇒ 8²= PC²+6²

⇒ PC²= 64–36 = 28

PC = 2√7 cm

Consider ΔACO,

AO²= OC²+AC² [By Pythagoras theorem]

⇒102 = OC2+62

⇒OC2 = 100−36 = 64

⇒OC = 8 cm
From the figure, OP = OC+PC = 8+2√7 cm.
Hence, the distance between the centres is (8+2√7) cm.

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What fraction of the whole circle is minor arc RP in the given figure ?
a)
1/5 of the circle
b)
1/2 of the circle
c)
1/4 of the circle
d)
1/3 of the circle
Correct answer is option 'D'. Can you explain this answer?

Zachary Foster answered

The central angle subtended by the minor arc RP is:
Minor arc RP =∠POQ =120^∘
The fraction of the circle represented by the minor arc RP is:
Fraction=Central Angle / Total Angle of the Circle
Substituting the values:
Fraction = 120^∘/ 360^∘= 1 / 3

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : The circumference of a circle must be a positive real number.
Reason : If r(>) 0 is the radius of the circle, then its circumference 2πr is a positive real number.
a)
Both assertion and reason are true and reason is the correct explanation of assertion.
b)
Both assertion and reason are true but reason is not the correct explanation of assertion.
c)
Assertion is true but reason is false.
d)
Assertion is false but reason is true
Correct answer is option 'A'. Can you explain this answer?

Aditi Sharma answered

The circumference of a circle is its perimeter or distance around it. It is denoted by C in math formulas and has units of distance, such as millimeters (mm), centimeters (cm), meters (m), or inches (in). It is related to the radius, diameter, and pi using the following equations: C = πd. C = 2πr.

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Arc ABC subtends an angle of 130^o at the centre O of the circle. AB is extended to P. Then ∠CBP equals :

a)
60^o

b)
70^o

c)
130^o

d)
65^o

Correct answer is option 'D'. Can you explain this answer?

EduRev Class 9 answered

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : In a cyclic quadrilateral ABCD, ∠A - ∠C = 60º, then the smaller of two is 60º.
Reason : Opposite angles of cyclic quadrilateral are supplementary.
a)
Both assertion and reason are true and reason is the correct explanation of assertion.
b)
Both assertion and reason are true but reason is not the correct explanation of assertion.
c)
Assertion is true but reason is false.
d)
Assertion is false but reason is true
Correct answer is option 'A'. Can you explain this answer?

Avinash Patel answered

Since ABCD is a cyclic quadrilateral, so, its opposite angles are supplementary

∠A + ∠C = 180º ...(1)

Also, ∠A - ∠C = 60º ...(2)

On solving (i) and (ii), we get

∠A = 120º, ∠C = 60º

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In the given figure, O is the centre of the circle and ∠AOC = 130^o. Then ∠ABC is equal to
a)
115^o
b)
130^o
c)
65^o
d)
165^o
Correct answer is option 'A'. Can you explain this answer?

Astha Shukla answered

Here is the answer

360 degree - 130 degree =230 degree

Therefore, angle ABC =1/2reflex angle AOC

Angle ABC =1/2 230 degree

So, angle ABC =115 degree

Hope it helps you

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : Given a circle of radius r and with centre O. A point P lies in a plane such that OP > r then point P lies on the exterior of the circle.
Reason : The region between an arc and the two radii, joining the centre of the end points of the arc, is called a sector.
a)
Both assertion and reason are true and reason is the correct explanation of assertion.
b)
Both assertion and reason are true but reason is not the correct explanation of assertion.
c)
Assertion is true but reason is false.
d)
Assertion is false but reason is true
Correct answer is option 'B'. Can you explain this answer?

Samarth Pillai answered

Assertion: Given a circle of radius r and with center O. A point P lies in a plane such that OP > r, then point P lies on the exterior of the circle.

Reason: The region between an arc and the two radii, joining the center of the endpoints of the arc, is called a sector.

To determine the correctness of the given assertion and reason, let's analyze each statement separately.

Assertion: Given a circle of radius r and with center O. A point P lies in a plane such that OP > r, then point P lies on the exterior of the circle.

This assertion is true. When the distance between the point P and the center O of the circle is greater than the radius r, it means that P is located outside the circle. This is because the radius is the distance from the center to any point on the circumference of the circle. If the distance between P and O is greater than the radius, P must be outside the circle.

Reason: The region between an arc and the two radii, joining the center of the endpoints of the arc, is called a sector.

This reason is true. A sector of a circle is a region enclosed by an arc and the two radii that connect the center of the circle to the endpoints of the arc. It is named based on the angle formed by the two radii at the center of the circle.

Now, let's evaluate the relationship between the assertion and reason.

The reason does not provide a direct explanation for the assertion. It discusses a different concept related to circles, which is the definition of a sector. Although the reason is true, it does not explain why a point P lying outside the circle would be represented by OP > r.

Therefore, the correct answer is option B: Both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : In an isosceles triangles ABC with AB = AC , a circle is passing through B and C intersects the sides AB and AC at D and E respectively. Then DE || BC
Reason : Exterior angle of a cyclic quadrilateral is equal to interior opposite angle of that quadrilateral
a)
Both assertion and reason are true and reason is the correct explanation of assertion.
b)
Both assertion and reason are true but reason is not the correct explanation of assertion.
c)
Assertion is true but reason is false.
d)
Assertion is false but reason is true.
Correct answer is option 'A'. Can you explain this answer?

Aditya Shah answered

To prove DE || BC

i.e., ∠B ∠ADE .

In Δ ABC , we have

AB = AC

∠B = ∠C ...(1)

In the cyclic quadrilateral CBDE , side BD is produced to A. We know that an exterior angle of cyclic quadrilateral is equal to interior opposite angle of cyclic quadrilateral.

∠ADE = ∠C ...(2)

From (1) and (2), we get

∠B = ∠ADE .

Hence, DE || BC

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Direction: In the following questions, a statement of Assertion is given followed by a corresponding statement of Reason just below it. Of the statements, mark the correct answer as
Assertion : The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
Reason : Two or more circles are called concentric circles if and only if they have different centre and radii.
a)
Both assertion and reason are true and reason is the correct explanation of assertion.
b)
Both assertion and reason are true but reason is not the correct explanation of assertion.
c)
Assertion is true but reason is false.
d)
Assertion is false but reason is true
Correct answer is option 'C'. Can you explain this answer?

Akshara Patel answered

Assertion: The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.

Reason: Two or more circles are called concentric circles if and only if they have different centre and radii.

Explanation:
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. In other words, the circumcircle of a cyclic quadrilateral passes through all four vertices of the quadrilateral. Let's analyze the given assertion and reason in detail:

Assertion: The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
In a cyclic quadrilateral, opposite angles are supplementary, which means they add up to 180º. This property can be proven using the fact that opposite angles in a cyclic quadrilateral are subtended by the same arc on the circumcircle.

Reason: Two or more circles are called concentric circles if and only if they have different centre and radii.
This reason is not directly related to the assertion and does not provide a correct explanation. The reason talks about concentric circles, which are circles that share the same center but have different radii. However, this concept is not relevant to the sum of opposite angles in a cyclic quadrilateral.

Conclusion:
The assertion is true because opposite angles in a cyclic quadrilateral are supplementary. However, the reason is false as it does not provide a correct explanation for the assertion. Hence, the correct answer is option 'C' - Assertion is true but reason is false.

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