Class 9 Mathematics Chapter 9 Circles

Key Concept: Angle Subtended by a Chord at a Point, Theorem 9.7

b) $40^\circ$

[Solution Description]

According to Theorem 9.7, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Here, the angle subtended at the centre O is $∠AOB = 80^\circ$. Therefore, the angle subtended at point C on the minor arc AB is $∠ACB = \frac{1}{2} \times ∠AOB = \frac{1}{2} \times 80^\circ = 40^\circ$.

Key Concept: Angle Subtended by a Chord at the Centre

c) θ = φ

[Solution Description]

According to Theorem 9.1, equal chords of a circle subtend equal angles at the centre. Therefore, if two chords are equal, the angles they subtend at the centre must also be equal. Hence, $θ = φ$.

Key Concept: Basic Concept

b) A portion of a circle enclosed by two radii and an arc

[Solution Description]

A sector is a portion of a circle enclosed by two radii and an arc.

Key Concept: Congruent Triangles Corresponding Angles, SSS Rule

c) $∠AOB = ∠COD$

[Solution Description]

Given that chords $AB$ and $CD$ are equal in length, and triangles $AOB$ and $COD$ are congruent, we can conclude that their corresponding angles are equal. According to the SSS rule, the sides of both triangles are equal: $OA = OC$ (radii of the circle), $OB = OD$ (radii of the circle), and $AB = CD$ (given). Therefore, $∠AOB = ∠COD$.

Key Concept: Arc and Angle at Centre

b) $30^\circ$

[Solution Description]

According to Theorem 9.7, the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Therefore, if the angle at the centre is $60^\circ$, the angle at any point on the remaining part of the circle is $\frac{1}{2} \times 60^\circ = 30^\circ$.

Key Concept: Diameter Calculation

b) 10 units

[Solution Description]

The circumference of a circle is given by the formula $C = \pi \times d$, where $d$ is the diameter. Rearranging the formula to solve for diameter:

$d = \frac{C}{\pi} = \frac{31.4}{3.14} = 10 \text{ units}$

Key Concept: Concept: Cyclic Quadrilateral

c) The quadrilateral is cyclic

[Solution Description] According to Theorem 9.11, if the sum of a pair of opposite angles of a quadrilateral is $180°$, the quadrilateral is cyclic. Therefore, since $∠A + ∠C = 180°$, the quadrilateral ABCD is cyclic.

Key Concept: Angle Subtended by a Chord

d) The angles are equal

[Solution Description]

When two chords of a circle are equal in length, they subtend equal angles at the center of the circle. This is because the angle subtended by a chord at the center is directly related to the length of the chord. Therefore, if the chords are equal, the angles they subtend at the center will also be equal.

Key Concept: Theorem 9.2

c) The chords are equal

[Solution Description] According to Theorem 9.2, if the angles subtended by the chords of a circle at the center are equal, then the chords are equal. Therefore, the correct conclusion is that the chords are equal.

Key Concept: Congruent Arcs

b) The arcs are congruent

[Solution Description]

According to the syllabus, if two chords of a circle are equal, their corresponding arcs are congruent. This means that one arc can superimpose the other completely without bending or twisting.

Key Concept: Theorem 9.4

c) 10 cm

[Solution Description]

Since OM is perpendicular to AB and M is the midpoint of AB, we can use the Pythagorean theorem in triangle OAM.

Given:

OM = 6 cm,

AM = 8 cm.

The radius OA is the hypotenuse of the right triangle OAM. Therefore,

$OA^2 = OM^2 + AM^2$

Substituting the given values:

$OA^2 = 6^2 + 8^2 = 36 + 64 = 100$

Taking the square root of both sides:

$OA = \sqrt{100} = 10 \text{ cm}$

So, the radius of the circle is 10 cm.

Key Concept: Theorem 9.7

b) $20^\circ$

[Solution Description]

According to Theorem 9.7, the angle subtended by an arc at the center is double the angle subtended at any point on the remaining part of the circle. Given that the angle at the center is $40^\circ$, the angle at the circumference will be half of this value.

Step-by-step solution:

1. We know that $\angle POQ = 2 \times \angle PAQ$.

2. Given $\angle POQ = 40^\circ$, we need to find $\angle PAQ$.

3. Therefore, $\angle PAQ = \frac{\angle POQ}{2} = \frac{40^\circ}{2} = 20^\circ$.

Key Concept: Basic Concept

b) 5 cm

[Solution Description]

The formula for the circumference of a circle is:

$C = 2\pi r$

Given that $C = 31.4$ cm and $\pi = 3.14$, we can solve for the radius $r$:

$31.4 = 2 \times 3.14 \times r$

Divide both sides by $2 \times 3.14$:

$r = \frac{31.4}{6.28} = 5\,\text{cm}$

Therefore, the radius is 5 cm.

Key Concept: Angle in a Semicircle

c) $90°$

[Solution Description]

The angle subtended by a diameter in a semicircle is always $90°$. This is a direct consequence of the theorem stating that any angle inscribed in a semicircle is a right angle.

Key Concept: Opposite Angles in Cyclic Quadrilateral

c) $180^\circ$

[Solution Description]

According to Theorem 9.10, the sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$.

Key Concept: Equal Chords and Angles

d) The angles are equal

[Solution Description]

According to Theorem 9.1, equal chords of a circle subtend equal angles at the centre. Therefore, if two chords are equal, the angles they subtend at the centre will also be equal.

Key Concept: Angles in the same segment

c) 30$^\circ$

[Solution Description]

According to Theorem 9.8, angles in the same segment of a circle are equal. The central angle $∠ COD$ is twice the inscribed angle $∠ CAD$. Therefore, $∠ CAD = \frac{1}{2} \times ∠ COD = \frac{1}{2} \times 60^\circ = 30^\circ$.

Key Concept: Theorem 9.2

c) The chords are equal

[Solution Description]

According to Theorem 9.2, if the angles subtended by the chords of a circle at the centre are equal, then the chords are equal in length. Therefore, if two chords subtend equal angles at the centre, they must be of equal length.

Key Concept: Chords Equidistant from the Centre

b) 8 cm

[Solution Description]

According to the properties of a circle, chords that are equidistant from the center are equal in length. Therefore, if AB is 8 cm long, then CD must also be 8 cm long.

Key Concept: Circle Theorem

b) On the line segment extended from the diameter

[Solution Description] According to the circle theorem, if two circles intersect at two points and a diameter is drawn through one of these points, the other intersection point will lie on the line segment formed by extending the diameter. This is because the angles subtended by the diameter at the other intersection point are right angles (90°), making the points colinear.

Key Concept: Congruent Circles and Chords

c) AB equals CD

[Solution Description]

According to Theorem 9.2, if the angles subtended by the chords of congruent circles at their centers are equal, then the chords are equal. Since both chords AB and CD subtend an angle of 60° at their respective centers, it follows that $AB = CD$.

Key Concept: Definition of a Cyclic Quadrilateral

d) It is cyclic.

[Solution Description]

According to Theorem 9.11, if the sum of a pair of opposite angles of a quadrilateral is $180°$, then the quadrilateral is cyclic. This means that all four vertices of the quadrilateral lie on a circle.

Key Concept: Theorem 9.2

b) 8 cm

[Solution Description]

Since both chords $XY$ and $YZ$ subtend equal angles ($45^\circ$) at the centre, by Theorem 9.2, the lengths of $XY$ and $YZ$ must be equal. Therefore, the length of $YZ$ is also 8 cm.

Key Concept: Definition of a Circle

b) A collection of points equidistant from a fixed point in a plane

[Solution Description]

A circle is defined as the collection of all points in a plane that are equidistant from a fixed point in the plane. This fixed point is called the center of the circle.

Key Concept: Cyclic Quadrilateral Opposite Angles

d) $110^\circ$

[Solution Description]

By Theorem 9.10, the sum of opposite angles in a cyclic quadrilateral is $180^\circ$. Given $\angle A = 70^\circ$, we can find $\angle C$ as follows:

$\angle A + \angle C = 180^\circ \\ \\ 70^\circ + \angle C = 180^\circ \\ \\ \angle C = 180^\circ – 70^\circ = 110^\circ$

Therefore, the measure of $\angle C$ is $110^\circ$.

Key Concept: Equal Chords and Their Distances from the Centre

c) 8 cm

[Solution Description] To find the distance of the chord from the center, we can use the formula:

$\text{Distance} = \sqrt{r^2 – \left(\frac{L}{2}\right)^2}$

where $r$ is the radius and $L$ is the length of the chord. Plugging in the values:

$\text{Distance} = \sqrt{10^2 – \left(\frac{12}{2}\right)^2} = \sqrt{100 – 36} = \sqrt{64} = 8 \text{ cm}$

Key Concept: Converse of Theorem 9.3

b) It is perpendicular to the chord

[Solution Description]

The converse of Theorem 9.3 states that if a line from the centre of a circle bisects a chord, then it is perpendicular to the chord. Therefore, the line is perpendicular to the chord.

Key Concept: Angle Subtended by Equal Chords at the Centre

c) The angles are equal

[Solution Description]

According to Theorem 9.1, equal chords of a circle subtend equal angles at the centre. Therefore, the angles subtended by two equal chords at the centre are equal.

Key Concept: Angle Subtended by a Chord

d) The angles are equal

[Solution Description] According to the syllabus, if two chords of a circle are equal in length, the angles they subtend at the center of the circle will also be equal. This is because the size of the angle subtended by a chord at the center is directly related to the length of the chord.

Key Concept: Equal Chords Subtend Equal Angles

d) The angles are equal

[Solution Description] According to Theorem 9.1, equal chords of a circle subtend equal angles at the centre. Therefore, if two chords are equal, the angles they subtend at the centre are also equal.

Key Concept: Angle Subtended by an Arc at the Centre

b) 20°

[Solution Description] According to **Theorem 9.7**, the angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle. Given that the angle at the centre is 40°, the angle at any point on the remaining part of the circle is half of 40°. Thus, $∠ PAQ = \frac{1}{2} × 40° = 20°$.

Key Concept: Angle in a Semicircle

c) $90^\circ$

[Solution Description]

According to the theorem, an angle in a semicircle is a right angle. Therefore, the measure of the angle is $90^\circ$.

Key Concept: Equal Chords and Their Distances from the Centre

c) They are equidistant

[Solution Description]

According to Theorem 9.5, equal chords of a circle are equidistant from the centre. Therefore, if two chords are equal in length, they must be equidistant from the centre of the circle.

Key Concept: Theorem 9.9

c) The points lie on a circle

[Solution Description]

According to Theorem 9.9, 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 (i.e. they are concyclic). Therefore, the points A, B, C, and D lie on a circle.

Key Concept: Cyclic Quadrilateral Angles

c) 100°

[Solution Description]

In a cyclic quadrilateral, the sum of opposite angles is $180°$. Given $∠ A = 80°$ and $∠ B = 100°$, we can find $∠ C$ as follows:

Since $∠ A + ∠ C = 180°$, substituting the value of $∠ A$ gives:

$80° + ∠ C = 180°$.

Therefore, $∠ C = 180° – 80° = 100°$.

Key Concept: Theorem 9.1, Theorem 9.2

c) AB and CD are equal in length

[Solution Description]

Since the circles are congruent, they have the same radii. According to Exercise 9.1, equal chords of congruent circles subtend equal angles at their centres. Given that both chords AB and CD subtend angles of $70^\circ$ at their respective centres, it follows that $AB = CD$.

Key Concept: Equal Chords and Their Distances from the Centre

c) AB and CD are equal in length

[Solution Description]

Since both chords AB and CD are at the same distance (5 cm) from the centre O, by Theorem 9.6, they must be equal in length. Hence, $AB = CD$.

Key Concept: Theorem 9.4

b) 8 cm

[Solution Description]

Step 1: Since the line passing through O bisects chord PQ, it divides the chord into two equal parts.

Step 2: Therefore, each segment of the chord will be half the length of PQ.

Step 3: Given that PQ is 16 cm, each segment will be 8 cm.

Key Concept: Equal Chords and Central Angles

d) They are equal

[Solution Description]

If two chords of a circle are equal in length, the angles they subtend at the center will also be equal. This is because the size of the central angle depends on the length of the chord.

Key Concept: Converse of Perpendicular from the Centre to a Chord

b) OM is perpendicular to AB

[Solution Description]

According to Theorem 9.4, the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. Since M is the mid-point of chord AB, the line OM will be perpendicular to AB.

Key Concept: Cyclic Quadrilateral Property

d) It is cyclic

[Solution Description]

According to Theorem 9.11, if the sum of a pair of opposite angles of a quadrilateral is $180^\circ$, then the quadrilateral is cyclic. This is the converse of Theorem 9.10.