1.17 Lengths and polygons in the coordinate plane
Remember that the coordinate plane can be used to describe the location of points in a 2D space.
By connecting 3 or more points on the coordinate plane with line segments, we can plot polygons. Plotting polygons on the coordinate plane will allow us to easily determine lengths and distances without needing a ruler.
A triangle on the coordinate plane
Later we will also look at how we can transform polygons on the coordinate plane.
Worked example
Question 1
Consider the point $Q$Q plotted on the coordinate plane.
a) 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).
Start at $Q$Q, then move $6$6 units right and $8$8 units down.
b) If point $R$R has the coordinates $\left(-4,-7\right)$(−4,−7), what is the distance between $Q$Q and $R$R?
Think: Notice that point $R$R has the same $x$x-coordinate as point $Q$Q. This means that the distance between the two points is given by the difference in the two $y$y-coordinates.
Do: So the distance is $3-\left(-7\right)=10$3−(−7)=10 units.
$Q$Q and $R$R have the same $x$x-coordinate.
Given a set of points, we may be asked what type of shape or polygon is formed. It can be very helpful to plot the points. From there you may need to calculate side lengths by counting squares or finding the difference in coordinates.
Practice questions
Question 2
What are the coordinates of the vertices of this quadrilateral?
$A\left(\editable{},\editable{}\right)$A(,), $B\left(\editable{},\editable{}\right)$B(,), $C\left(\editable{},\editable{}\right)$C(,), $D\left(\editable{},\editable{}\right)$D(,)
Question 3
What is the distance between $A\left(-6,2\right)$A(−6,2) and $B\left(-6,-7\right)$B(−6,−7)?