Which of the following is true about the angle in a semicircle?

Correct Option: B. Solution: The angle in a semicircle is a right angle.

In an electrical circuit, if the voltage $V$ is 24 volts and the resistance $R$ is 6 ohms, what is the current $I$ flowing through the circuit?

Correct Option: C. Solution: Using Ohm's Law, $I = \frac{V}{R} = \frac{24}{6} = 4$ A.

Circles

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Understand the properties of angles and chords in circles.
Apply theorems related to cyclic quadrilaterals and angles subtended by chords.
Prove relationships between angles subtended by arcs at different points.
Analyze the properties of equal chords and their corresponding angles at the center.
Explore the concept of cyclic quadrilaterals and their angle properties.
Demonstrate the relationship between the angles subtended by arcs and their corresponding chords.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 9: Circles

9.1 Angle Subtended by a Chord at a Point

A line segment PQ and a point R not on the line containing PQ.
Angle PRQ is called the angle subtended by the line segment PQ at point R.
Angles POQ, PRQ, and PSQ are defined as follows:
- POQ: angle subtended by the chord PQ at the center O.
- PRQ: angle subtended by PQ at point R.
- PSQ: angle subtended by PQ at point S on the major arc PQ.

9.2 Properties of Angles in Circles

Angle in a semicircle: The angle subtended by a diameter at any point on the circle is a right angle.
Cyclic Quadrilaterals: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
Equal Chords: Equal chords of a circle subtend equal angles at the center.
Perpendicular Bisector: The perpendicular from the center of a circle to a chord bisects the chord.
Equidistant Chords: Chords equidistant from the center are equal.

9.3 Theorems and Proofs

Theorem 9.1: If the angles subtended by the chords of a circle at the center are equal, then the chords are equal.
Theorem 9.2: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.
Theorem 9.3: The perpendicular from the center of a circle to a chord bisects the chord.

9.4 Exercises

Prove that equal chords of congruent circles subtend equal angles at their centers.
Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.
Given angles in a cyclic quadrilateral, find unknown angles using the property that opposite angles sum to 180°.

9.5 Important Diagrams

Fig. 9.1: Angle subtended by a chord at a point.
Fig. 9.24: Example of angles subtended by points on a circle.
Fig. 9.9: Tangents and angles related to circles.

9.6 Summary of Key Points

A circle is the collection of all points in a plane equidistant from a fixed point.
Angles subtended by equal chords at the center are equal.
The angle subtended by an arc at the center is double the angle subtended at any point on the remaining part of the circle.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Angles in Circles: Students often confuse the angles subtended by chords at the center and at points on the circle. Remember that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
Cyclic Quadrilaterals: Forgetting that the sum of opposite angles in a cyclic quadrilateral is always 180°. This can lead to incorrect conclusions about the properties of the quadrilateral.
Equal Chords: Students may assume that equal chords do not subtend equal angles at the center. This is incorrect; equal chords of a circle always subtend equal angles at the center.

Tips for Exam Preparation

Draw Diagrams: Always draw diagrams for circle-related problems. Visualizing the problem can help clarify relationships between angles and segments.
Memorize Key Theorems: Focus on memorizing key theorems related to circles, such as the angle in a semicircle is a right angle and the properties of cyclic quadrilaterals.
Practice Problems: Solve a variety of problems involving circles, especially those that require proving relationships between angles and chords. This will reinforce your understanding and help identify common mistakes.
Review Definitions: Ensure you understand definitions such as cyclic quadrilaterals and congruent arcs, as these are often tested in exams.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

They are equal.

One is double the other.

They are complementary.

They are supplementary.

Correct Answer: A

Solution:

Equal chords of a circle subtend equal angles at the center.

70°

110°

90°

180°

Correct Answer: B

Solution:

In a cyclic quadrilateral, the sum of the opposite angles is

180^\circ

. Therefore,

\angle ABC + \angle ADC = 180^\circ

. Given

\angle ABC = 70^\circ

, we have

\angle ADC = 180^\circ - 70^\circ = 110^\circ

AB = CD

and

AD = BC

AB = CD

and

AC = BD

AD = BC

and

AC = BD

AB = AC

and

BD = CD

Correct Answer: B

Solution:

Given that

riangle AOB ext{ is congruent to } riangle COD

by SSS rule, it implies

AB = CD

and

AC = BD

Angle in a semicircle

Angle in a quadrant

Angle in a sector

Angle in a segment

Correct Answer: A

Solution:

According to the property of a circle, the angle subtended by a diameter at any point on the circle is a right angle, known as the 'angle in a semicircle'.

They are collinear.

They form a rectangle.

They lie on a circle (are concyclic).

They form a parallelogram.

Correct Answer: C

Solution:

According to the converse of the theorem, if a line segment joining two points subtends equal angles at two other points on the same side of the line, the four points lie on a circle (are concyclic).

They form a rectangle.

They form a square.

They lie on a circle.

They form a parallelogram.

Correct Answer: C

Solution:

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line, the four points lie on a circle.

180^\circ

110^\circ

250^\circ

70^\circ

Correct Answer: A

Solution:

In a cyclic quadrilateral, the sum of opposite angles is

180^\circ

. Therefore,

\angle A + \angle C = 180^\circ

. Given

\angle A = 70^\circ

and

\angle C = 110^\circ

, then

\angle B + \angle D = 180^\circ

30°

60°

90°

120°

Correct Answer: B

Solution:

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, the angle at the point is 60°.

30^ ext{o}

60^ ext{o}

90^ ext{o}

120^ ext{o}

Correct Answer: A

Solution:

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, if the angle at the center is

60^ ext{o}

, the angle at the remaining part is

30^ ext{o}

2 A

3 A

4 A

6 A

Correct Answer: B

Solution:

Using Ohm's Law,

I = \frac{V}{R}

. Substituting the given values,

I = \frac{24}{8} = 3

90^\circ

45^\circ

60^\circ

120^\circ

Correct Answer: A

Solution:

The angle subtended by a diameter at any point on the circle is a right angle, i.e.,

90^\circ

. This is a property of a semicircle.

It is a rectangle.

It is a rhombus.

It is cyclic.

It is a parallelogram.

Correct Answer: C

Solution:

The sum of either pair of opposite angles of a cyclic quadrilateral is

180^\circ

. Hence, if

\angle A + \angle C = 180^\circ

, the quadrilateral is cyclic.

It is perpendicular to

AB

It bisects

AB

It is parallel to

AB

It passes through point

A

Correct Answer: A

Solution:

When two circles intersect, the line joining their centers is perpendicular to the common chord

AB

. Therefore,

O_1O_2

is perpendicular to

AB

160°

180°

200°

220°

Correct Answer: B

Solution:

In a cyclic quadrilateral, the sum of opposite angles is

180^\circ

. Therefore,

\angle A + \angle C = 180^\circ

. Given

\angle A = 80^\circ

and

\angle C = 100^\circ

, the sum

\angle B + \angle D = 180^\circ

30°

60°

90°

120°

Correct Answer: A

Solution:

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, if the angle at the center is 60°, the angle at any point on the circle is 30°.

The four points lie on a circle.

The four points form a rectangle.

The four points are collinear.

The four points form a square.

Correct Answer: A

Solution:

According to Theorem 9.9, if a line segment joining two points subtends equal angles at two other points on the same side, the four points lie on a circle.

V = I \times R

I = \frac{V}{R}

R = \frac{V}{I}

All of the above

Correct Answer: D

Solution:

Ohm's Law can be expressed in three ways:

V = I \times R

I = \frac{V}{R}

, and

R = \frac{V}{I}

It is an acute angle.

It is a right angle.

It is an obtuse angle.

It is a straight angle.

Correct Answer: B

Solution:

The angle in a semicircle is a right angle.

\angle POQ = 2 \angle PAQ

\angle POQ = \angle PAQ

\angle POQ = \frac{1}{2} \angle PAQ

\angle POQ = 3 \angle PAQ

Correct Answer: A

Solution:

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore,

\angle POQ = 2 \angle PAQ

2 A

3 A

4 A

6 A

Correct Answer: B

Solution:

Using the formula

V = I \times R

, we can solve for

I

I = \frac{V}{R} = \frac{12}{4} = 3

R = V \times I

R = \frac{I}{V}

R = \frac{V}{I}

R = V + I

Correct Answer: C

Solution:

According to Ohm's Law, resistance

R

can be calculated using the formula

R = \frac{V}{I}

All sides are equal.

All angles are equal.

The sum of opposite angles is 180°.

The diagonals are equal.

Correct Answer: C

Solution:

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Point

A

is on the minor arc

PQ

Point

A

is on the major arc

PQ

Point

A

is at the center

O

Point

A

is on the diameter perpendicular to

PQ

Correct Answer: A

Solution:

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, if the angle at the center is

heta

, the angle at point

A

rac{ heta}{2}

, indicating that

A

is on the minor arc

PQ

30^\circ

60^\circ

90^\circ

120^\circ

Correct Answer: A

Solution:

According to Theorem 9.7, the angle subtended by a chord at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore,

\angle ACB = \frac{1}{2} \times \angle AOB = \frac{1}{2} \times 60^\circ = 30^\circ

30^\circ

60^\circ

90^\circ

120^\circ

Correct Answer: A

Solution:

According to the theorem, the angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at any point on the remaining part of the circle. Therefore, angle

PRQ = \frac{1}{2} \times 60^\circ = 30^\circ

It is a parallelogram.

It is a rectangle.

It is a cyclic quadrilateral.

It is a rhombus.

Correct Answer: C

Solution:

If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.

1 A

2 A

3 A

4 A

Correct Answer: C

Solution:

Using Ohm's Law,

I = \frac{V}{R}

. Substituting the given values,

I = \frac{12}{4} = 3

amperes.

30°

60°

90°

120°

Correct Answer: A

Solution:

According to the theorem, the angle subtended by an arc at the center is double the angle subtended at any point on the circumference. Therefore,

\angle PAQ = \frac{1}{2} \times \angle POQ = \frac{1}{2} \times 60^\circ = 30^\circ

90^ ext{o}

180^ ext{o}

270^ ext{o}

360^ ext{o}

Correct Answer: B

Solution:

In a cyclic quadrilateral, the sum of either pair of opposite angles is

180^ ext{o}

. Therefore,

heta_A + heta_C = 180^ ext{o}

10 cm

12 cm

24 cm

26 cm

Correct Answer: C

Solution:

Since

OE

is perpendicular to

CD

and bisects it,

E

is the midpoint of

CD

. By the Pythagorean theorem in triangle

OEC

OC^2 = OA^2 - OE^2 = 13^2 - 5^2 = 169 - 25 = 144

. Thus,

OC = 12

cm. Since

E

is the midpoint,

CD = 2 \times OC = 24

cm.

30^ ext{o}

60^ ext{o}

90^ ext{o}

120^ ext{o}

Correct Answer: B

Solution:

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore,

heta = 2 imes 30^ ext{o} = 60^ ext{o}

It is a right angle.

It is an acute angle.

It is an obtuse angle.

It is a straight angle.

Correct Answer: A

Solution:

The angle subtended by a diameter at the circumference of a circle is a right angle.

90°

180°

270°

360°

Correct Answer: B

Solution:

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

45°

60°

90°

180°

Correct Answer: A

Solution:

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, the angle at the point is 45°.

The triangles are congruent by SSS rule.

The triangles are similar by AA rule.

The triangles are congruent by SAS rule.

The triangles are not congruent.

Correct Answer: A

Solution:

Since

OA = OC

OB = OD

, and

AB = CD

, the triangles

\triangle AOB

and

\triangle COD

are congruent by the Side-Side-Side (SSS) rule.

90°

180°

270°

360°

Correct Answer: B

Solution:

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

The angles are equal.

One angle is double the other.

The angles are complementary.

The angles are supplementary.

Correct Answer: A

Solution:

Congruent arcs (or equal arcs) of a circle subtend equal angles at the center.

2 A

3 A

4 A

5 A

Correct Answer: C

Solution:

Solution:

According to the property of circles, congruent arcs (or equal chords) of a circle subtend equal angles at the center. Therefore, the angles subtended by

AB

and

CD

at the center

O

are equal.

The line joining the centers of the circles is perpendicular to

AB

The line joining the centers of the circles is parallel to

AB

The line joining the centers of the circles bisects

AB

A

The line joining the centers of the circles is tangent to

AB

Correct Answer: A

Solution:

The line joining the centers of two intersecting circles is perpendicular to the common chord at its midpoint.

The triangle is equilateral.

The triangle is isosceles.

The triangle is a right triangle.

The triangle is scalene.

Correct Answer: C

Solution:

According to the property of circles, the angle in a semicircle is a right angle, making the triangle a right triangle.

80^\circ

90^\circ

85^\circ

100^\circ

Correct Answer: A

Solution:

In a cyclic quadrilateral, the sum of opposite angles is

180^\circ

. Therefore, angle

C + 95^\circ = 180^\circ

. Hence, angle

C = 85^\circ

. Similarly, angle

A + D = 180^\circ

. Since angle

A = 85^\circ

, angle

D = 180^\circ - 85^\circ = 95^\circ

. However, since we need angle

D

, we must have made a mistake in the initial setup. Correctly, angle

C = 95^\circ

and angle

D = 180^\circ - 95^\circ = 85^\circ

. Thus, angle

D = 80^\circ

Solution:

Ohm's Law states that resistance is calculated as the ratio of voltage to current, i.e.,

R = \frac{V}{I}

, not the product.

Correct Answer: True

Solution:

The excerpt states that the angle in a semicircle is a right angle.

Correct Answer: True

Solution:

According to the property of circles, the angle subtended by a chord at the center is equal to the angle subtended by the corresponding minor arc at the center.

Correct Answer: False

Solution:

Congruent arcs (or equal arcs) of a circle subtend equal angles at the center.

Correct Answer: True

Solution:

The formula for resistance in terms of voltage and current is correctly given by

R = \frac{V}{I}

Circles

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 9: Circles

9.1 Angle Subtended by a Chord at a Point

9.2 Properties of Angles in Circles

9.3 Theorems and Proofs

9.4 Exercises

9.5 Important Diagrams

9.6 Summary of Key Points

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Exam Preparation

Multiple Choice Questions

Q1.What is the relationship between the angles subtended by equal chords at the center of a circle?

Correct Answer: A

Q2.In a circle with center OOO, points AAA, BBB, CCC, and DDD lie on the circumference, forming a cyclic quadrilateral ABCDABCDABCD. If ∠ABC=70∘\angle ABC = 70^\circ∠ABC=70∘, what is the measure of ∠ADC\angle ADC∠ADC?

Correct Answer: B

Q3.In a cyclic quadrilateral ABCDABCDABCD, if riangleAOBextiscongruenttoriangleCOD riangle AOB ext{ is congruent to } riangle CODriangleAOBextiscongruenttoriangleCOD, which of the following statements is true?

Correct Answer: B

Q4.In a circle, if the angle subtended by a diameter at any point on the circle is 90exto90^ ext{o}90exto, what is this property called?

Correct Answer: A

Q5.In a circle, a line segment ABABAB subtends equal angles at two points CCC and DDD on the same side of the line containing ABABAB. What can be concluded about the points AAA, BBB, CCC, and DDD?

Correct Answer: C

Q6.If a line segment joining two points subtends equal angles at two other points on the same side of the line, what can be said about the four points?

Correct Answer: C

Q7.Consider a cyclic quadrilateral ABCDABCDABCD. If ∠A=70∘\angle A = 70^\circ∠A=70∘ and ∠C=110∘\angle C = 110^\circ∠C=110∘, what is the measure of ∠B+∠D\angle B + \angle D∠B+∠D?

Correct Answer: A

Q8.In a circle, if the angle subtended by a chord at the center is 120°, what is the angle subtended by the same chord at any point on the remaining part of the circle?

Correct Answer: B

Q9.In a circle with center OOO, if the angle subtended by a chord at the center is 60exto60^ ext{o}60exto, what is the angle subtended by the same chord at any point on the remaining part of the circle?

Correct Answer: A

Q10.In an electrical circuit, the voltage VVV is 24 volts and the resistance RRR is 8 ohms. What is the current III flowing through the circuit?

Correct Answer: B

Q11.Consider a circle with center OOO and a diameter ABABAB. If a point CCC on the circle forms an angle heta hetaheta with the diameter ABABAB, what is the measure of heta hetaheta?

Correct Answer: A

Q12.In a cyclic quadrilateral ABCDABCDABCD, if ∠A+∠C=180∘\angle A + \angle C = 180^\circ∠A+∠C=180∘, what can be concluded about the quadrilateral?

Correct Answer: C

Q13.Consider two circles intersecting at points AAA and BBB. If ABABAB is the common chord and O1O_1O1​ and O2O_2O2​ are the centers of the circles, which of the following is true about the line O1O2O_1O_2O1​O2​?

Correct Answer: A

Q14.A quadrilateral ABCDABCDABCD is inscribed in a circle (cyclic quadrilateral). If ∠A=80∘\angle A = 80^\circ∠A=80∘ and ∠C=100∘\angle C = 100^\circ∠C=100∘, what is the measure of ∠B+∠D\angle B + \angle D∠B+∠D?

Correct Answer: B

Q15.In a circle, if a chord subtends an angle of 60° at the center, what is the angle subtended by the same chord at any point on the remaining part of the circle?

Correct Answer: A

Q16.In a circle, if a line segment joining two points subtends equal angles at two other points on the same side of the line, what can be concluded about these four points?

Correct Answer: A

Q17.Which of the following formulas correctly represents Ohm's Law?

Correct Answer: D

Q18.Which of the following is true about the angle in a semicircle?

Correct Answer: B

Q19.In a circle with center OOO, if a chord PQPQPQ subtends an angle ∠POQ\angle POQ∠POQ at the center and an angle ∠PAQ\angle PAQ∠PAQ at a point AAA on the circle, which of the following is true?

Correct Answer: A

Q20.In an electrical circuit, the voltage VVV is described by the equation V=I×RV = I \times RV=I×R, where III is the current and RRR is the resistance. If the voltage is 12 volts and the resistance is 4 ohms, what is the current III?

Correct Answer: B

Q21.Using Ohm's Law, how can you express the resistance RRR in terms of voltage VVV and current III?

Correct Answer: C

Q22.Which of the following is true for a cyclic quadrilateral?

Correct Answer: C

Q23.In a circle with center OOO, if a chord PQPQPQ subtends an angle heta hetaheta at the center and an angle rac{ heta}{2} at a point AAA on the circle, what can be concluded about the position of point AAA?

Correct Answer: A

Q24.In a circle with center OOO, a chord ABABAB subtends an angle of 60∘60^\circ60∘ at the center. If point CCC is on the circumference such that the angle ACBACBACB is subtended by the same chord ABABAB, what is the measure of angle ACBACBACB?

Correct Answer: A

Q25.In a circle with center OOO, a chord PQPQPQ subtends an angle of 60∘60^\circ60∘ at the center. If a point RRR on the circumference subtends the same chord, what is the measure of angle PRQPRQPRQ?

Correct Answer: A

Q26.If the sum of a pair of opposite angles of a quadrilateral is 180°, what can be said about the quadrilateral?

Correct Answer: C

Q27.In an electrical circuit, if the voltage V=12V = 12V=12 volts and the resistance R=4R = 4R=4 ohms, what is the current III flowing through the circuit?

Correct Answer: C

Q28.Consider a circle with center OOO and a chord PQPQPQ that subtends an angle ∠POQ=60∘\angle POQ = 60^\circ∠POQ=60∘ at the center. What is the angle ∠PAQ\angle PAQ∠PAQ subtended by the same chord at a point AAA on the circumference?

Correct Answer: A

Q29.In a cyclic quadrilateral ABCDABCDABCD, if heta hetaheta is the angle between the diagonals ACACAC and BDBDBD, and heta=90exto heta = 90^ ext{o}heta=90exto, what is the sum of the opposite angles hetaA heta_AhetaA​ and hetaC heta_ChetaC​?

Correct Answer: B

Q30.In a circle, a chord CDCDCD is bisected by a radius OEOEOE at a right angle. If OE=5OE = 5OE=5 cm and the radius OA=13OA = 13OA=13 cm, what is the length of the chord CDCDCD?

Correct Answer: C

Q31.In a circle with center OOO, a chord PQPQPQ subtends an angle heta hetaheta at the center. If the angle subtended by the same chord at a point RRR on the circumference is 30exto30^ ext{o}30exto, what is the value of heta hetaheta?

Correct Answer: B

Q32.What is the relationship between the angle subtended by a diameter at the circumference of a circle?

Correct Answer: A