Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 ↓


Coordinate Geometry


Introduction

Coordinate geometry, also sometimes known as analytical geometry, combines algebra and geometry to describe the position and relationships of figures in a plane. This branch of mathematics relies on a coordinate system to uniquely identify the location of points. This allows us to use algebraic methods to solve geometric problems.

Cartesian plane

The most common coordinate system used in coordinate geometry is the Cartesian plane. It consists of two number lines that are perpendicular to each other and intersect at their zero points. These number lines are called axes. The horizontal line is the x-axis, and the vertical line is the y-axis.

The point where the x-axis and y-axis meet is called the origin, represented by the coordinates (0, 0). Every point in the plane is associated with an ordered pair of numbers, often called its coordinates.

Example: Understanding the Cartesian plane

Consider the following depiction of the Cartesian plane:

(0,0) (1,1)

In this diagram, the red circle with coordinates (1, 1) is located one unit to the right and one unit up from the origin. Each move to the right increases the x-coordinate and each move up increases the y-coordinate.

Plotting points

To plot a point on the Cartesian plane you need to know its coordinates (x, y).

  • The x-coordinate, or abscissa, represents the position of the point along the x-axis. Positive values indicate positions to the right of the origin, while negative values indicate positions to the left.
  • The y-coordinate represents the position of the point along the y-axis. Positive values place the point above the origin, while negative values place it below it.

Example: Plotting points

For example, let's plot the points (2, 3), (-1, -2) and (0, -4):

(2, 3) (-1, -2) (0, -4)

In this visual representation:

  • The point (2, 3) is 2 units to the right of the y-axis and 3 units above the x-axis.
  • The point (-1, -2) is 1 unit to the left of the y-axis and 2 units below the x-axis.
  • The point (0, -4) lies exactly on the y-axis and 4 units below the x-axis.

Quadrant

The Cartesian plane is divided into four quadrants by the x-axis and y-axis:

  1. Quadrant I: Both the x and y coordinates are positive.
  2. Quadrant II: x is negative and y is positive.
  3. Quadrant III: Both the x and y coordinates are negative.
  4. Fourth quadrant: x is positive and y is negative.
Quadrant I Quadrant II Quadrant III fourth quadrant

Distance formula

The distance between any two points in the Cartesian plane can be calculated using the distance formula. Suppose we have two points: A (x1, y1) and B (x2, y2). The distance d between these points can be calculated as follows:

d = √((x2 - x1)² + (y2 - y1)²)

Example: Finding distance

Let us find the distance between the points (3, 4) and (7, 1):

Let A = (3, 4) and B = (7, 1)
d = √((7 - 3)² + (1 - 4)²) = √(4² + (-3)²) = √(16 + 9) = √25 = 5

The distance between these two points is 5 units.

Midpoint formula

The midpoint between two points on the Cartesian plane can be easily calculated. Suppose we have two points: point A (x1, y1) and point B (x2, y2), the midpoint M is given by:

M = ((x1 + x2) / 2, (y1 + y2) / 2)

Example: Calculating the midpoint

Consider the points (1, 2) and (-3, 4). Let us find their midpoint:

Let A = (1, 2) and B = (-3, 4)
M = ((1 + (-3)) / 2, (2 + 4) / 2) = (-2 / 2, 6 / 2) = (-1, 3)

The midpoint is at (-1, 3).

Equation of line

One of the major results of coordinate geometry is the ability to find the equation of a line. The most common form of the equation of a line is the slope-intercept form:

y = mx + c
  • m represents the slope of the line, which indicates its depth.
  • c is the y-intercept, where the line intersects the y-axis.

The second form is the point-slope form, which is particularly useful when we know the points on the line and its slope:

y - y1 = m(x - x1)

where (x1, y1) is a known point on the line.

Example: Writing the equation of a line

Suppose we know that a line passes through the point (2, 3) and has a slope of 4. Using the point-slope formula:

y - 3 = 4(x - 2)

Expanding this equation gives:

y = 4x - 8 + 3 = 4x - 5

The equation of the line is y = 4x - 5.

Slope of the line

The slope of a line measures its inclination to the horizontal. It is calculated as the ratio of the vertical change to the horizontal change between two points on the line:

m = (y2 - y1) / (x2 - x1)

Example: Calculating slope

Consider the points (6, 2) and (8, 6):

m = (6 - 2) / (8 - 6) = 4 / 2 = 2

The slope of the line passing through these points is 2.

Applications of coordinate geometry

Coordinate geometry has a lot of applications ranging from graphical representation of shapes to solving real-world problems involving distances, trajectories, and more. Many engineering, architecture, and physics problems rely heavily on the principles of coordinate geometry.

Conclusion

Coordinate geometry is the bridge between algebra and geometry, facilitating the understanding of space and shape properties through numerical and analytical approaches. It equips students with the tools needed to explore mathematical concepts and solve practical challenges, laying a foundation for more advanced mathematics and allied fields.

As you progress in your mathematical studies, mastering coordinate geometry will strengthen your problem-solving skills, making it easier to understand more complex topics in your future learning journey.


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