Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Circles ↓


Chord Properties


A chord is a straight line that joins two points on the circumference of a circle. Among the various properties and concepts related to circles, chords play an important role, especially in geometry. Understanding the properties of chords gives us the ability to explore a wide range of geometric concepts and solve complex problems involving circles. Let us delve deeper into the properties of chords and understand them in detail.

Basic properties of chords

Before we explore chord properties further, let's consider a basic understanding of what a circle is. A circle is a closed figure where all points are the same distance from a center point. This constant distance from the center to the circumference is called the radius.

Now, consider a chord in a circle. The line segment forming the chord divides the circle into two distinct regions, which can be segments or arcs. Some basic properties of a chord are:

  • The longest chord of a circle is its diameter.
  • If the chords are the same length then they are equidistant from the center of the circle.

Equal chords and distance from the centre

An important property of chords is the concept of equidistance from the center. In other words, if two chords are equal in length, then they are equidistant from the center. Conversely, if two chords are equidistant from the center, then they must be equal in length.

For example, consider a circle with center (O). If the chords (AB) and (CD) are equal, then they are equidistant from (O). Let's represent this visually in the following example:


    
        
        
        
        Hey
        A
        B
        C
        D

Perpendicular from the center to the chord

If you draw a perpendicular from the center of a circle to a chord, this perpendicular bisects the chord. This is another important property of chords in circles.

This property implies that if (OM) is the perpendicular from the center (O) to the chord (AB), then (M) is the midpoint of (AB). Let's see this with a visual aid:


    
        
        
        
        
        Hey
        A
        B
        M

Chord development and arch

There is an interesting relationship between a chord and the arcs it forms on a circle. Two main theories deal with this relationship:

  1. If two chords of a circle are equal, then the arcs subtended by them will also be equal.
  2. Conversely, if two arcs are equal, then the chords intersecting them are also equal.

For example, if ( overset{frown}{AB} ) and ( overset{frown}{CD} ) are equal arcs, then the chords (AB) and (CD) are also equal in length. Let's explain this:


    
        
        
        
        B
        A
        D
        C

Angles in the same segment

The angles subtended by the chord at any point on the remaining segment of the circle are equal. This principle is known as the angles in equal segments theorem.

Consider a chord (AB). Now, draw two points (C) and (D) on the circumference, which lie on the same side of the chord. Then, (angle ACB = angle ADB). Let's visualize this concept:


    
        
        
        
        
        
        A
        B
        D
        C

Perpendicular bisector of a chord

The perpendicular bisector of a chord passes through the center of the circle. This property is constructive when it comes to drawing the center of a circle using chords.

Let us take a chord (AB) and its perpendicular bisector (CD). According to this property, the line (CD) will cut the chord (AB) at its midpoint and pass through the center of the circle. Note:


    
        
        
        
        
        Hey
        A
        B
        D
        C

Examples and exercises

Let's apply these properties of chords and solve some exercises. This will give you a comprehensive understanding of the concepts.

Example 1

Given two equal chords (AB) and (CD) in a circle with centre (O), prove that:

  • The perpendicular drawn from (O) bisects (AB) and (CD)
  • The angles in the segments formed by (AB) and (CD) are equal

Solution: Since the chords are equal ((AB = CD)), they are equidistant from (O). Therefore, (OM) and (ON) are perpendicular bisectors of (AB) and (CD), respectively.

Example 2

A chord (AB) of length 8 cm, perpendicular distance from the center (O) is 3 cm is given by:

  • Calculate the radius of the circle.

Solution: Let the perpendicular drawn from (O) meets (AB) at (M). Thus, (M) is 4 cm, since the chord (AB) is bisected:

According to the Pythagorean theorem:

OM 2 + AM 2 = OA 2 3 2 + 4 2 = OA 2 9 + 16 = OA 2 OA = 5 cm (radius)

Exercise

Try solving these problems to solidify your understanding:

  1. If two chords of a circle are equal, then prove that the distances of these chords from the centre of the circle are equal.
  2. In a circle of radius 10 cm, a chord is 12 cm long. Find the distance of the chord from the centre of the circle.

Conclusion

The study of chords within circles provides us with a deeper understanding of the geometry of circles. Understanding the relationships and properties associated with chords is fundamental in solving geometric problems in more advanced topics. Through visualization and practical application, the concept of chords can become less abstract and more intuitive, serving as the foundational framework for a myriad of geometric constructions and theorems.


Grade 9 → 10.3


U
username
0%
completed in Grade 9

← Prev (10.2)
Properties of a Circle
Next (10.4) →
Tangents to a Circle

Comments 



Chord Properties

https://www.buddymath.com/g9/10.3?bmal=en