Rectangular Coordinate System in a Plane

The use of rectangular coordinate system is presented along with examples, questions and their solutions.

Rectangular Coordinate System in a Plane

A rectangular coordinate system in a Plane is used to plot points having an \( x \) coordinate and a \( y \) coordinate. A vertical number line, also called the y-axis, and a horizontal number line, also called x-axis, intersecting at a right angle form a system of coordinates in a plane as shown in figure 1 below. The point of intersection of the x and y axes is called the origin of the system of coordinates.
The x and y axes split the plane into four quadrants as noted in the figure above.


Cartesian Plane
Fig.1 - Rectangular Coordinate System in a Plane
Note that the rectangular coordinate system is also called the Cartesian coordinate system .



Point Plotting in Rectangualar Coordinate System

Each point in the plane corresponds to an ordered pair \( (x,y) \) , where \( x \) and \( y \) are real numbers. \( x \) and \( y \) are called the coordinates of the point where the \( x \) coordinate represents the directed distance from the y axis to the point and the \( y \) coordinate represent the directed distance from the x-axis to the point.
Example
In the example below, the \( x \) coordinate of point \( A \) is \( 4 \) and therefore positive, hence point \( A \) is located to the right of the y-axis in the direction of the arrow of the x-axis.
The \( x \) coordinate of point \( B \) is \( -5 \) and therefore negative, hence point \( B \) is located to the left of the y-axis in the direction opposite to the arrow of the x-axis..
The \( y \) coordinate of point \( A \) is \( 2 \) and therefore positive, hence point \( A \) is located above the x-axis in the direction of the arrow of the y-axis..
The \( y \) coordinate of point \( B \) is \( -2 \) and therefore negative, hence point \( B \) is located above the x-axis in the direction opposite to the arrow of the y-axis..

Point Plotting in a Rectangular Coordinate System  in a Plane
Fig.2 - Example of Point Plotting in a Rectangular Coordinate System
More practice on plotting points in rectangular coordinate system is included.



Signs of Coordinates and Quadrants

The signs of the \( x \) and \( y \) coordinates of a given point, gives enough information to find the quadrant of that point without plotting it.
Example
Points \( A = (4,2) \) and \( C = (-4,3) \) are both located above the x axis because their \( y \) coordinates \( 2 \) and \( 3 \) are both positive. But point \( A \) is on the right of the y axis and hence in quadrant I because its \(x \) coordinate \( 4 \) is positive.
The \( x \) coordinate of \( C \), which is \( -4 \), is negative and therefore point \( C \) is to the left of the y axis, hence located in quadrant II.
Similar remarks could be made about points \( E = (-4,-2) \) in quadrant III and \( G = (4,-2) \) in quadrant IV.
Conclusion
Given a point with coordinates \( x \) and \( y \), and without plotting the point, we can find the quadrant where the point will be located from the signs of \( x \) and \( y \).
If \( x \gt 0 \) and \( y \gt 0 \), the point is in quadrant I
If \( x \lt 0 \) and \( y \gt 0 \), the point is in quadrant II
If \( x \lt 0 \) and \( y \lt 0 \), the point is in quadrant III
If \( x \gt 0 \) and \( y \lt 0 \), the point is in quadrant IV

Signs of coorcdinates in the different quadrant
Fig.3 - Signs of the Coordinates in the Four Different Quadrants of a Rectangular System



Point on the x and y Axes

Any point whose \( x \) coordinate is equal to zero, is located in the y axis because its distance from the y axis is equal to zero.
Example
Points \( A = (0,3) \), \( D = (0,-2) \) and \( E = (0,-4) \) all have the \( x \) coordinate equal to zero and are therefore located in the y axis. (see figure 4 below)

Any point whose \( y \) coordinate is equal to zero, is located in the x axis because its distance from the x axis is equal to zero.
Example
Points \( G = (6,0) \), \( C = (-2,0) \) and \( B = (-8,0) \) all have the \( y \) coordinate equal to zero and are therefore located in the x axis. (see figure 4 below)

Point on x and y axes of a Rectangular Coordinate System
Fig.4 - Point on the x and y axes



Questions

Part A

Plot the following points: \( A = (0,0) \; , \; B = (-4,3) \; , \; C = (0,-4) \; ; \; D = (5,-5) \; ; \; E = (-3,0) \; ; \; F = (-2,-3) \; ; \; G = (4,0) \; ; \; H = (2,5) \)

Part B

Give the coordinates of all points plotted in the graph below.
Point on a Rectangular Coordinate System

Part C

Without plotting the points given below; in which quadrant or axis is each of the following points located?
\( A = (-32,-89) \; , \; B = (0,45) \; , \; C = (-88,0) \; ; \; D = (57,89) \; ; \; E = (0,-77) \; ; \; F = (45,-38) \; ; \; G = (49,0) \; ; \; H = (-90,-56) \)

Part D

Graph each of the following group of points, link the points in the order that they are given and describe each quadrilateral obtained.
Group 1: \( A = (2,2) \; , \; B = (-4,2) \; , \; C = (-4,-1) \; , \; D = (2,-1) \)
Group 2: \( A = (1,2) \; , \; B = (-2,2) \; , \; C = (-5,-2) \; , \; D = (5,-2) \)
Group 3: \( A = (0,4) \; , \; B = (-2,2) \; , \; C = (0,-4) \; , \; D = (2,2) \)



Solutions to the Above Questions

Part A

Point on a Rectangular Coordinate System for part A

Part B

\( A = (0,1) \), \( B = (2,0) \), \( C = (1,3) \), \( D = (-1,-1) \), \( E = (1,-3) \), \( F = (-3,1) \), \( G = (0,-4) \), \( H = (-3,0) \)

Part C

\( A = (-32,-89) \) in quadrant III
\( B = (0,45) \) on the y axis, above the x axis
\( C = (-88,0) \) on the x axis, to the left the y axis
\( D = (57,89) \) in quadrant I
\( E = (0,-77) \) on the y axis, below the x axis
\( F = (45,-38) \) in quadrant IV
\( G = (49,0) \) on the x axis, to the right the y axis
\( H = (-90,-56) \) in quadrant III

Part D

Group 1: \( A = (2,2) \; , \; B = (-4,2) \; , \; C = (-4,-1) \; , \; D = (2,-1) \)
The four given points form a rectangle as shown below.
Points forming a rectangle

Group 2: \( A = (1,2) \; , \; B = (-2,2) \; , \; C = (-5,-2) \; , \; D = (5,-2) \)
The four given points form a trapezoid as shown below.
Points forming a trapezoid

Group 3: \( A = (0,4) \; , \; B = (-2,2) \; , \; C = (0,-4) \; , \; D = (2,2) \)
The four given points form a kite as shown below.
Points forming a kite



More References and Links

plotting points in rectangular coordinate system
Geometry Tutorials and Problems
The Four Pillars of Geometry - John Stillwell - Springer; 2005th edition (Aug. 9 2005) - ISBN-10 : 0387255303
Geometry: A Comprehensive Course - Daniel Pedoe - Dover Publications - 2013 - ISBN: 9780486131733
Geometry: with Geometry Explorer - Michael Hvidsten - McGraw Hill - 2006 - ISBN: 0-07-294863-9