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

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

Rectangular Coordinate System in a Plane