Circles

Simplified Quantitative Formulas: Circles

Circumference: 2πr. Area: πr².
Diameter: Subtends right angle at circumference.
Equal Chords: Equal arcs, angles, and distance from center.
Tangent: Perpendicular to radius at point of contact.
Tangents from Point: Equal in length.
Perpendicular from Center: Bisects chord.
Inscribed Angle Theorem: Angle on circle = ½ angle at center.
Cyclic Quadrilateral: Opposite angles sum to 180°.
Segment Area: Minor = sector – triangle area.

1. Introduction

Understanding Circles

A circle is a set of all points in a plane that are equidistant from a fixed point called the center.

Basic Properties:

All points on the circle are equidistant from the center
The distance from center to any point is called radius
Diameter is twice the radius
Circumference = 2πr
Area = πr²

Basic Circle Elements

2. Key Properties & Formulas

General Equation of a Circle:

Standard form: x² + y² + 2gx + 2fy + c = 0
Centre: (-g, -f)
Radius: √(g² + f² - c)
Origin as centre: x² + y² = r²

Complementary angles: Sum to 90°
Supplementary angles: Sum to 180°
Intersecting lines: Opposite (vertical) angles are equal; adjacent angles are supplementary
Sum of angles at a point: 360°
Angle subtended by diameter: Always 90° (right angle)
Angles subtended by equal chords: Are equal; angles in the major segment are half the angle at the center
Equal chords: Are equidistant from the center
Radius to tangent: Perpendicular at the point of contact
Tangents from the same point: Are equal in length
Perpendicular from center to chord: Bisects the chord
Area of sector OAXC: (θ/360) × πr²
Area of minor segment AXC: (θ/360) × πr² - (1/2)r²sinθ
Inscribed angle theorem: Angle at center is twice the angle at the circle (2∠ACB = ∠AOB)
Angles subtended by same segment: Are equal
Angle between chord and tangent: Equals angle subtended by chord in the alternate segment (∠ACB = ∠BAT)
Power of a point: AP × AQ = AS × AU = AT² (for tangents and secants from A)
Direct common tangent (between circles): PQ² = RS² = D² - (r₁ - r₂)², where D = distance between centers
Transverse common tangent: PQ² = RS² = D² - (r₁ + r₂)²

2. Basic Elements

Circle Components

Arc and Chord

Sector and Segment

Formulas:

Arc Length = (θ/360°) × 2πr
Sector Area = (θ/360°) × πr²
Segment Area = Sector Area - Triangle Area
Chord Length = 2r × sin(θ/2)

Practice Questions - Basic Elements

Question 1

A circle has a radius of 7cm. Find its circumference and area.

Solution:

1. Circumference = 2πr = 2 × π × 7 = 14π cm
2. Area = πr² = π × 7² = 49π cm²

Question 2

A chord of length 8cm is 3cm away from the center of a circle. Find the radius of the circle.

Solution:

1. Using Pythagoras theorem:
2. r² = 4² + 3² (half chord length = 4cm)
3. r² = 16 + 9 = 25
4. r = 5cm

Question 3

Find the length of an arc that subtends an angle of 60° at the center of a circle with radius 10cm.

Solution:

1. Arc length = (θ/360°) × 2πr
2. = (60/360) × 2π × 10
3. = (1/6) × 20π
4. = 10π/3 cm

Question 4

A sector of a circle has an area of 25π cm² and a central angle of 90°. Find the radius of the circle.

Solution:

1. Sector area = (θ/360°) × πr²
2. 25π = (90/360) × πr²
3. 25π = (1/4) × πr²
4. r² = 100
5. r = 10cm

3. Circle Theorems

Key Theorems with Proofs

1. Inscribed Angle Theorem

Inscribed angle = (1/2) × Central angle

Proof:

Draw radii to the points of the inscribed angle
This creates isosceles triangles
Use angle sum property of triangles
Prove that inscribed angle is half the central angle

Inscribed Angle Theorem

2. Cyclic Quadrilateral Theorem

Opposite angles sum to 180°

Proof:

Draw the quadrilateral inscribed in a circle
Use the Inscribed Angle Theorem
Show that opposite angles intercept opposite arcs
Prove that their sum is 180°

Cyclic Quadrilateral

3. Tangent-Secant Theorem

PT² = PA × PB

Proof:

Draw tangent PT and secant PAB
Use Power of a Point theorem
Show that triangles are similar
Prove the relationship PT² = PA × PB

Tangent-Secant Theorem

Practice Questions - Theorems

Question 1

In a circle, an inscribed angle intercepts an arc of 120°. Find the measure of the inscribed angle.

Solution:

1. Using Inscribed Angle Theorem:
2. Inscribed angle = (1/2) × Central angle
3. = (1/2) × 120°
4. = 60°

Question 2

In a cyclic quadrilateral ABCD, angle A = 70° and angle B = 110°. Find angles C and D.

Solution:

1. Using Cyclic Quadrilateral Theorem:
2. Opposite angles sum to 180°
3. If A = 70°, then C = 110°
4. If B = 110°, then D = 70°

Question 3

A tangent PT and a secant PAB are drawn to a circle. If PA = 4cm and PB = 9cm, find the length of PT.

Solution:

1. Using Tangent-Secant Theorem:
2. PT² = PA × PB
3. PT² = 4 × 9 = 36
4. PT = 6cm

Question 4

In a circle, two chords AB and CD intersect at point P. If AP = 3cm, PB = 4cm, and CP = 6cm, find PD.

Solution:

1. Using Intersecting Chords Theorem:
2. AP × PB = CP × PD
3. 3 × 4 = 6 × PD
4. 12 = 6 × PD
5. PD = 2cm

4. Tangents & Secants

Properties of Tangents

Tangent Properties:

Tangent is perpendicular to radius at point of contact
Two tangents from external point are equal in length
Angle between tangent and chord equals angle in alternate segment

Tangent Properties

Practice Questions - Tangents

Question 1

Two tangents are drawn from an external point P to a circle. If the angle between the tangents is 60°, find the angle between the radii drawn to the points of contact.

Solution:

1. The angle between radii = 180° - angle between tangents
2. = 180° - 60°
3. = 120°

Question 2

A tangent is drawn to a circle of radius 5cm from a point 13cm away from the center. Find the length of the tangent.

Solution:

1. Using Pythagoras theorem:
2. Tangent length² = Distance² - Radius²
3. = 13² - 5²
4. = 169 - 25 = 144
5. Tangent length = 12cm

Question 3

In a circle, a tangent and a chord meet at a point. If the angle between the tangent and chord is 40°, find the angle subtended by the chord in the alternate segment.

Solution:

1. Using Alternate Segment Theorem:
2. Angle in alternate segment = Angle between tangent and chord
3. = 40°

Question 4

Two circles touch externally. The distance between their centers is 10cm and the sum of their radii is 10cm. Find the radii of the circles.

Solution:

1. Let radii be r₁ and r₂
2. r₁ + r₂ = 10cm (given)
3. Distance between centers = r₁ + r₂ = 10cm
4. Therefore, r₁ = r₂ = 5cm

5. Applications

Real-World Applications

Common Applications:

Wheel and gear design
Architecture and construction
Navigation and surveying
Sports and games

Example 1

Find the length of a tangent drawn from a point 5cm away from the center of a circle with radius 3cm.

Solution:

1. Using Pythagoras theorem:
2. Tangent length² = Distance² - Radius²
3. Tangent length² = 5² - 3²
4. Tangent length² = 25 - 9 = 16
5. Tangent length = 4cm

Practice Questions - Applications

Question 1

A wheel of radius 35cm makes 100 revolutions to cover a certain distance. Find the distance covered.

Solution:

1. Circumference = 2πr = 2π × 35 = 70π cm
2. Distance = Circumference × Number of revolutions
3. = 70π × 100
4. = 7000π cm

Question 2

A circular garden has a diameter of 14m. A path of width 2m is to be laid around it. Find the area of the path.

Solution:

1. Garden radius = 7m
2. Outer radius = 7 + 2 = 9m
3. Path area = π(9² - 7²)
4. = π(81 - 49)
5. = 32π m²

Question 3

A circular track has an inner radius of 100m and outer radius of 105m. Find the area of the track.

Solution:

1. Track area = π(R² - r²)
2. = π(105² - 100²)
3. = π(11025 - 10000)
4. = 1025π m²

Question 4

A circular swimming pool has a diameter of 10m. A concrete path of width 1m is to be laid around it. Find the cost of laying the path at ₹100 per square meter.

Solution:

1. Pool radius = 5m
2. Outer radius = 6m
3. Path area = π(6² - 5²)
4. = π(36 - 25) = 11π m²
5. Cost = 11π × 100 = ₹1100π

Table of Contents

Simplified Quantitative Formulas: Circles

1. Introduction

Understanding Circles

2. Key Properties & Formulas

2. Basic Elements

Circle Components

Practice Questions - Basic Elements

3. Circle Theorems

Key Theorems with Proofs

Practice Questions - Theorems

4. Tangents & Secants

Properties of Tangents

Practice Questions - Tangents

5. Applications

Real-World Applications

Practice Questions - Applications

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