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 ↓


Properties of a Circle


The circle is a simple yet fascinating geometric shape, and it is defined as the set of all points in a plane that are at a fixed distance from a given point, called the center. Let us take a deeper look at the various properties of a circle to understand it better.

Center and radius

The center of a circle is the point from which every point on the circle is equidistant. This distance from the center to any point on the circle is called the radius. If we denote the center of the circle by C and any point on the circle by P, then the circle can be defined using the following equation:

(x - h)² + (y - k)² = r²

Here, (h, k) is the center of the circle, and r is the radius.

C P R

Diameter

The diameter of a circle is twice the radius. It is the longest distance passing through the centre of the circle. The diameter can be expressed as:

d = 2r

where d is the diameter and r is the radius.

D

Circumference

The circumference of a circle is the distance around it. It can be calculated using the formula:

C = 2πr

Alternatively, it can be expressed using diameter as:

C = πd

Where π (pi) is approximately 3.14159.

Area

The area enclosed by a circle is given by:

A = πr²

This formula helps in calculating the amount of space occupied by a circle.

Wire

A chord is a line segment whose both ends are on a circle. The longest chord in a circle is its diameter.

Wire

Arch

An arc is a part of the circumference of a circle. An arc can be measured in degrees. A semicircle is an arc that represents half of a circle.

Arch

Area

A sector is an area bounded by two radii and an arc. It looks like a slice of pie.

Area

Tangent line

A tangent is a line that touches a circle at exactly one point. It never goes inside the circle.

tangent line

Understanding the properties of a circle with examples

Let's look at some examples to understand these properties better:

Example 1: Finding the diameter

Suppose the radius of a circle is 7 cm. To find the diameter of the circle:

d = 2r = 2 × 7 = 14 cm

Example 2: Calculating perimeter

If the radius of a circle is 5 cm, then the circumference can be calculated as follows:

C = 2πr = 2 × 3.14159 × 5 ≈ 31.4159 cm

Example 3: Area of a circle

The area of a circle of radius 10 cm is:

A = πr² = 3.14159 × 100 = 314.159 cm²

Example 4: Understanding chords

A chord of a circle of radius 5 cm is 8 cm long. Can you find how far the chord is from the centre?

Using the perpendicular from the centre to the chord:

Let the distance from center to chord be h . By Pythagorean theorem: r² = (chord length/2)² + h² 5² = 4² + h² 25 = 16 + h² h² = 25 - 16 h² = 9 h = 3 cm

Example 5: Tangent line property

The tangent to a circle and the radius drawn at the tangent point are perpendicular. So a tangent at point P to a circle with center C is given by:

CP ⊥ Tangent at P

These examples highlight the versatility and applications of the properties of a circle. Understanding these properties can be interesting because they lay the groundwork for advanced geometry topics.


Grade 9 → 10.2


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Properties of a Circle

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