Number Plane and Cartesian Geometry
Theory, Formulae and Practice
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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
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.
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
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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:
Distance and Midpoint
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