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 ↓


Cartesian Plane


The Cartesian plane is a fundamental concept in coordinate geometry, which is an essential part of mathematics. It is a two-dimensional plane formed by the intersection of a vertical line called the y-axis and a horizontal line called the x-axis. These two axes divide the plane into four quadrants. Each point on the Cartesian plane is specified by a pair of numerical coordinates, which are the distances to the point from two intersecting, perpendicular axes.

Understanding the axes

The Cartesian plane is made up of two number lines that run perpendicular to each other:

  • x-axis (horizontal line): A line that extends from left to right. Positive numbers lie to the right of the origin, and negative numbers lie to the left.
  • y-axis (vertical line): A line that goes from top to bottom. Positive numbers lie above the origin, while negative numbers lie below.
+y -y +X -X Origin (0,0)

Coordinates

Every point on the plane is represented by a pair of numbers enclosed in brackets: (x, y). Here, x is the distance from the y-axis, and y is the distance from the x-axis. These numbers are known as coordinates. In mathematics, they are called the x-coordinate or abscissa and the y-coordinate or ordinate, respectively.

For example, the point (3, 2) lies 3 units to the right of the y-axis and 2 units above the x-axis. Conversely, the point (-4, -3) lies 4 units to the left of the y-axis and 2 units above the x-axis. It is located 3 units below the axis.

Four quadrants

The Cartesian plane is divided into four quadrants:

Quadrant I (+,+) Quadrant II (-,+) Quadrant III (-,-) Fourth Quadrant (+,-)
  1. Quadrant I: Both x and y coordinates are positive. Example: (3, 4)
  2. Quadrant II: x is negative, and y is positive. Example: (-3, 4)
  3. Quadrant III: Both x and y coordinates are negative. Example: (-3, -4)
  4. Fourth quadrant: x is positive, and y is negative. Example: (3, -4)

Drawing points on the Cartesian plane

To plot a point on the Cartesian plane, you start at the origin (0,0). Then, move horizontally to the x value of the point and vertically to the y value.

Let's plot the point (4, 3):

  1. Start at the origin (0,0).
  2. Move 4 units to the right along the x-axis.
  3. Move 3 units up along the y-axis.
(4, 3) Origin (0,0)

Produce

The point where the x-axis and y-axis intersect is called the origin. Its coordinates are (0, 0). The origin is the reference point for all other points on the plane.

Applications of Cartesian plane

The Cartesian plane has many practical applications in daily life as well as in advanced science.

Mathematics

In mathematics, the Cartesian plane is used to visualize and solve equations. For example, the equation of a straight line y = mx + b can be graphed on the Cartesian plane, where m is the slope and b is the perpendicular to the y-axis intersection.

Real-life examples

  • Geography: Mapping places using latitude and longitude.
  • Architecture: Designing buildings and structures with precise dimensions.
  • Navigation: GPS systems use coordinate geometry to locate positions.

Lines and curves on the Cartesian plane

The Cartesian plane can also be used to plot complex shapes and analyze geometric figures. Consider some of the following examples:

Linear equations

Linear equations such as y = 2x + 1 can be represented explicitly on the Cartesian plane. Here is a plot of such a line:

Curved lines

Not all equations will yield straight lines. Curves such as circles and parabolas are also usually represented on the Cartesian plane.

Solving problems

Let's solve a simple problem using the Cartesian plane.

Example problem

You are given two points A(3, 4) and B(7, 8). Find the midpoint of the line segment joining these points.

Solution

To find the midpoint M(x, y) between two points A(x1, y1) and B(x2, y2) use the formula:

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

Substitute the given values:

M(x, y) = ((3 + 7)/2, (4 + 8)/2) = (5, 6)

Thus, the midpoint is (5, 6).

Conclusion

Understanding the Cartesian plane is important for anyone studying coordinate geometry. The plane is not only a tool for visualizing mathematical concepts, but is also essential in many practical applications. As you learn to plot points and create coordinate graphs, you'll need to understand the Cartesian plane. As you become more comfortable with interpreting numbers, you will find that many mathematical problems become more accessible and solvable.


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Cartesian Plane

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