A grid reference system is commonly used on maps and makes use of combinations of letters and numbers to locate different positions within an area.
In mathematics, the Cartesian plane (also called the number plane, or coordinate plane) is used to describe the location of points, and provide a means to graphically represent mathematical relationships in a two-dimensional space.
A number plane is made up of a horizontal and a vertical axis.
The $x$x-axis is numbered with negative numbers, zero and positive numbers increasing to the right.
The $y$y-axis is numbered with negative numbers, zero and positive numbers increasing vertically.
Points are labelled on the Cartesian plane by an ordered pair of numbers of the form $\left(x,y\right)$(x,y), called coordinates.
The point at which the $x$x-axis and the $y$y-axis intercept is called the origin. The coordinates of the origin are $\left(0,0\right)$(0,0).
All other points on the Cartesian plane are labelled according to where a vertical and horizontal line, drawn from the point, intercept the $x$x-and $y$y axes respectively. For example, consider the point $Q$Q plotted on the number plane.
To find the $x$x-coordinate we can draw a vertical line from $Q$Q and read off the number at the point where this line intercepts the $x$x-axis (the horizontal axis). This gives us the number $-4$−4. Similarly, we can find the $y$y-coordinate by drawing a horizontal line from $Q$Q and reading the number at the point where this line intercepts the $y$y-axis (the vertical axis), which gives the number $3$3. So the coordinates of $Q$Q are $\left(-4,3\right)$(−4,3).
Notice the order of this process. We draw a vertical line toward the horizontal axis to find the $x$x-coordinate, and we draw a horizontal line toward the vertical axis to find the $y$y-coordinate.
In the following questions, we will identify coordinates in the first quadrant.
What are the coordinates of the marked point?
$\left(7,6\right)$(7,6)
$6,7$6,7
$7,6$7,6
$\left(6,7\right)$(6,7)
State the coordinates of the marked point.
$\left(\editable{},\editable{}\right)$(,)
What are the coordinates of the point shown in the number plane?
$\left(\editable{},\editable{}\right)$(,)
The Cartesian plane has four distinct regions, called quadrants. The applet below shows how these quadrants are defined.
Notice that the 1st quadrant is at the top-right of the Cartesian plane. The $x$x-coordinate and $y$y-coordinate of a point in the 1st quadrant are both positive.
Moving around anticlockwise we cover the other three quadrants, which have the following features:
Points that lie on an axis, like $\left(-5,0\right)$(−5,0) or $\left(0,4\right)$(0,4), are not considered to be in any quadrant.
(a) What quadrant is $Q$Q in?
Think: Earlier in the lesson we found that $Q$Q has a negative $x$x-coordinate and a positive $y$y-coordinate.
Do: This means that it is in the 2nd quadrant, in the top left of the number plane.
(b) What are the coordinates of the point that is $6$6 units to the right and $8$8 units below $Q$Q?
Think: Starting at point $Q$Q with coordinates $\left(-4,3\right)$(−4,3), we add $6$6 units to the $x$x-coordinate (because we are moving to the right), and subtract $8$8 units from the $y$y-coordinate (because we are moving downward).
Do: This gives $\left(-4+6,3-8\right)=\left(2,-5\right)$(−4+6,3−8)=(2,−5).
Consider the point $\left(5,0\right)$(5,0).
Plot the point on the number plane.
Where on the number plane is the point $\left(5,0\right)$(5,0) located?
2nd quadrant
No quadrant
3rd quadrant
1st quadrant
4th quadrant
In which quadrant does the point $\left(3,-2\right)$(3,−2) lie?
1st quadrant
2nd quadrant
3rd quadrant
4th quadrant
Consider the points $A\left(-4,8\right)$A(−4,8), $B\left(-7,8\right)$B(−7,8) and $C\left(-7,1\right)$C(−7,1).
Plot the points on the number plane.
What is the length of $AB$AB?
What is the length of $BC$BC?
Write the coordinates of the point that is $7$7 units to the right of $\left(-1,-2\right)$(−1,−2).
Coordinates $=$=$\left(\editable{},\editable{}\right)$(,)
What is the distance between $A\left(-6,2\right)$A(−6,2) and $B\left(-6,-7\right)$B(−6,−7)?