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Number Plane and Cartesian Geometry

Theory, Formulae and Practice

Welcome to Sydney Based Aussie Math Tutor NSW, your go-to destination for expert private math tutoring online. On this page you will learn the basic of graph in a clear easy way. The page is made by our experienced tutors in Sydney for the students in Sydney. Thus, master concepts such plotting the points, quadrants and plotting a line equation in this lesson.

CARTESIAN PLANE

A Number Plane is a flat-surfaced two-dimensional grid that extends infinitely and is formed by the intersection of two number lines. One number line runs horizontally and is called the X-axis. The other number line runs vertically and is called the Y-axis. These lines intersect at a point called the origin and is denoted as (0,0).
Quadrants: The four sections of the Cartesian plane divided by the x-axis and y-axis. The X-axis and the Y-axis divides the Number plane into four sections called as the Quadrants. The x and the y values of any point maybe positive or negative depending on which quadrant it lies.
The image below shows the number plane along with the Quadrants
Number Plane
Number Plane

The Co-ordinates of a point on a Number Plane

The Co-ordinates of any point is determined by its distance from the X-axis and the Y-axis. The x co-ordinate is the distance of the point away from the Y-axis and the y co-ordinate is the distance of the point away from the X-axis.
To find these distances, we draw a segment perpendicular to the X-axis and the Y-axis as they are the Shortest Distance to the Axes.
The image below shows how to find x and y co-ordinates of a give point.
Co-ordinates of a point on a Number Plane
Co-ordinates of a point on a Number Plane

Exercise: Practice Questions on Plotting the Points

Question : Plot the following points on a graph:
a) ( 0, 3 ) b) ( 1, 2 ) c) ( 2, 1 )
d) ( 3, -4 ) e) ( -4, -4 ) f) ( -5, 3 )
Answers for plaotting points on a graph
Answers for plaotting points on a graph

Answers

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Equation of a Line

The most common form of the equation of a line is called as the slope-intercept form: y = mx + c
Where:
(x,y) are the co-ordinates of a point on the graph
m is the slope of the line. The slope represents the steepness of the line.
c is the constant. The constant represents the distance away from the centre O (0,0).
The formula for the slope of a line is given by:
Slope of a Line
Slope of a Line

Distance and Midpoint

Distance and Midpoint between two points
Distance and Midpoint between two points

Relationship Between Slope and Parallel or Perpendicular Lines

Parallel Lines

  • Definition: Two lines are parallel if they have the same slope and never intersect.
  • Slope Relationship:
    • If two lines are parallel, their slopes are equal.
    • If line 1 has a slope m1 and line 2 has a slope m2, then for the lines to be parallel: m1=m2​

Perpendicular Lines

  • Definition: Two lines are perpendicular if they intersect at a right angle (90 degrees).
  • Slope Relationship:
    • If two lines are perpendicular, the product of their slopes is −1
    • If line 1 has a slope m1​ and line 2 has a slope m2, then for the lines to be perpendicular: m1×m2=−1

Drawing a Line from a Linear Equation

The most common way to draw a line from a linear equation is by converting the equation in the slope-intercept form: y = mx + c.
Substitute different values of x into the equation to obtain corresponding y values, creating at least two points. While plotting more points can increase the accuracy of the graph, only two points are necessary to draw a straight line representing the equation."
Two Points Suffice: While more points can improve accuracy, two points are sufficient to draw a straight line. All points on the line will satisfy the original equation.

Example on how to Draw a Line from a Linear Equation

Question : Draw the line for the equation x + y =3
Step 1: Convert the equation in the slope-intercept form: y = mx + c. Hence, the equation become, y= - x + 3.
Step 2: Substitute different values of x into the equation to obtain corresponding y values. Hence,
When x = -1 , y = 4. Therefor the point on the line is ( -1, 4 ).
When x = 0 , y = 3. Therefor the point on the line is ( 0, 3 ).
When x = 1 , y = 2. Therefor the point on the line is ( 1, 2 ).
When x = 2 , y = 1. Therefor the point on the line is ( 2, 1 ).
Step 3: Plot the points on the graph:
From the above graph, it can be seen that all the points lie on in a single line. Hence, The points are collinear points.
Step 4: Draw a line connecting all the points. Finish naming the graph, lines and the points.
Plotting points on a graph
Plotting points on a graph
Line of the Equation x+y=3
Line of the Equation x+y=3