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 ↓


Definitions and Terms


In geometry, a circle is a simple closed figure formed by all the points in the plane that are at a fixed distance from a given point, the center. The distance of any point on the circle from the center is called the radius. The circle is a fundamental figure in the study of geometry and it provides insight into various mathematical concepts. We will explore the basic terminology and definitions related to circles, provide a clear understanding and illustrate these concepts with examples.

Definition of circle

Mathematically a circle is defined as the set of all points in a plane that are at a certain fixed distance from a certain fixed point. This fixed point is called the center of the circle, and the fixed distance is called the radius. If the center of a circle is at the point (h, k) and the radius is r, then all points (x, y) on the circle satisfy the following relation:

    (x - h)^2 + (y - k)^2 = r^2

Components of a circle

  1. Center: The central point of a circle, from which all points on the circumference are equidistant. In the Cartesian coordinate system, the location of the center can be expressed as (h, k).
  2. Radius: A line segment from the center of a circle to any point on the circle. The radius is always perpendicular to the tangent to the circle at the endpoint of the circle.
  3. Diameter: A chord that passes through the center of a circle. It is the longest distance of the circle and is equal to twice the radius. i.e. Diameter = 2 * Radius.
  4. Circumference: Circumference refers to the entire distance around the circle. For a circle with radius r, the circumference C is given by the formula: 
     c = 2 * π * r
    where π is approximately 3.14159.
  5. Arc: A portion of the circumference of a circle. When you draw two distinct points on a circle and connect them by the shortest path along the circle, the resulting segment is called an arc.
  6. Chord: A line segment whose endpoints lie on a circle. The diameter is the longest chord of a circle.
  7. Secant: A line that intersects a circle at two points. Unlike a chord, a secant is not limited to the points of a circle but extends indefinitely in the plane.
  8. Tangent: A line that touches a circle at exactly one point. The tangent is perpendicular to the radius at the point of contact.
radius Diameter tangent line

In the visual example above, the black circle represents a standard circle with labeled elements. The red line represents the diameter passing through the center. The green line segment from the center to the circumference is the radius. The blue line is a tangent, touching the circle at a single point.

Properties of circle

Understanding the properties of a circle is very important in solving geometric problems. Some of the key properties include:

  • All the radii of a circle are equal.
  • The longest chord of a circle is its diameter.
  • The angle subtended by the diameter at any point on a circle is a right angle (90 degrees).
  • Dividing the circumference of any circle by its diameter gives the constant π.

Let's look at a few more examples to understand how these properties work in practice. Imagine you have a circle with a center at (5, 5) and a radius of 5 The equation of the circle is:

    (x - 5)^2 + (y - 5)^2 = 5^2

Examples and problem solving

Let us use some practical examples to understand the applications of these concepts.

Example 1: Calculating perimeter

If the radius of a circle is 7 cm, find the circumference.

To solve this, use the perimeter formula:

    c = 2 * π * r

Substitute the radius value:

    c = 2 * π * 7 ≈ 2 * 3.14159 * 7 ≈ 43.98 cm

Hence the circumference of the circle is approximately 43.98 cm.

Example 2: Finding the equation of a circle

Suppose a circle has centre (1, 2) and radius 4 units. Write the equation of the circle.

Plug the values into the standard circle equation:

    (x - 1)^2 + (y - 2)^2 = 4^2

This makes it simpler:

    (x - 1)^2 + (y - 2)^2 = 16

Therefore, the equation of the circle is (x - 1)^2 + (y - 2)^2 = 16.

Example 3: Working with arc

Given a circle of radius 10 cm, what is the length of the arc subtending an angle of 60 degrees at the centre?

Use the formula for arc length:

    Arc length = (θ / 360) * 2 * π * r

Here, θ = 60 degrees and r = 10 cm. Substitute these values:

    Arc length = (60/360) * 2 * π * 10 ≈ (1/6) * 2 * 3.14159 * 10 ≈ 10.47 cm

Hence the length of the arc is approximately 10.47 cm.

Example 4: Identifying the tangent

Consider a circle with centre (0, 0) and radius 5 If a line passes through the point (5, 0) and is perpendicular to the radius at that point, show that this line is a tangent.

Since the line is perpendicular to the radius, it touches the circle at exactly one point, which is the defining characteristic of a tangent line. Thus, the line through the point (5, 0) that meets the circle perpendicularly is the tangent line.

radius tangent line

This diagram shows how the radius is perpendicular to the tangent at the point (5, 0).

Conclusion

Understanding the terms and definitions related to circles is very important to understand more complex topics in geometry. With these definitions, terminology and examples, you can better understand and apply geometric principles and solve related problems efficiently. Remember to practice these principles with additional exercises, deepening your understanding of concepts such as radius, diameter, circumference, arc, chord, secant and tangent, all of which are fundamental to the geometry of circles.


Grade 9 → 10.1


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Definitions and Terms

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