Using Coordinates in Geometry

Although we often like to think of algebra and geometry as different subjects, they are very much related and being able to use algebra in geometry (and vice versa) can be very helpful when it comes to solving more advanced problems!

As we know from algebra, coordinates (x,y) are points on a plane.  Two points (two sets of coordinates) make up a line (and have a slope, remember: $m=\dfrac{y_2-y_1}{x_2-x_1}$?).

Coordinate geometry (which involves lots of lines, with slopes, often grouped into figures) brings together what we know about coordinates in equations and applies them to figures in a coordinate plate.

A space defined by an x-axis and a y-axis is a coordinate plane.

There are several rules that apply to all coordinate planes.

• x-coordinates are positive when they are to the right of the y-axis and negative to the left of the y-axis.
• y-coordinates are positive above the x-axis and negative below the x-axis.
• The origin is the place where the x- and y-axes cross (0,0).
• Coordinate pairs are always listed with the $x$ first and the $y$ second $(x,y)$. (It's alphabetical!)
• You can subtract $x$ coordinates to find the horizontal distance between two points.
• You can subtract $y$ coordinates to find the vertical distance between two points.
• You can use the distance formula to find the diagonal distance between two points: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Some coordinate geometry problems require you to use actual points. These problems are almost always easiest when you draw them out!

Example: There are two points on a coordinate plane, point A (3,2) and point B (-3,2). What is the distance between the two points.

Students who know the distance formula might immediately start to use it here: distance formula will give you the distance between two points!

But, if you draw these points, you see that they are on on a horizontal line.

Once you see that, it's easy to find the distance: either count the units on the graph you drew, or $3-(-3)=6$.

The points are 6 units apart.

Note: if this had been a diagonal line, and you did not remember the distance formula, you could create a right triangle and use Pythagorean Theorem to find the length of the diagonal line.

As questions get harder, they require more drawing, but students should continue to use their logic, even as problems start to seem very hard (even impossible!).

Example: Four points make a rectangle: (4, -1), (-2, -1), (4, $h$), and (-2,6).  What is the value of $h$?

This problem sounds tricky. But, once you draw it out, you see that the coordinates of the each point are obvious, because rectangles have all right angles.

Plot all the given points (see red points).

Figure out where the last dot MUST go (green point).

The last point is (4, 6) so $h=6$.

Sometimes the most difficult coordinate geometry questions don't require precise math at all.  They just require students to understand how numbers change as they move around the coordinate plane.

Example: In the figure below, if $\overline{AB}$ is a line segment, Point E is at (0,0), and $AE=EB$, what are the coordinates of B in terms of $x$ and $y$?

In this case, you are not meant to figure out the exact value of $x$ and $y$ or the exact coordinates of B, just how the coordinates of A relate to the coordinates of B.

First, make an educated guess about the coordinates of A.

Since you don't really need to know exactly what they are, pick easy numbers.  Just make sure to get the positive and negative signs right!

$x$ is to the left of the y-axis, so it must be negative.  Let's estimate $x\approx-5$

$y$ above the x-axis, so it must be positive.  Let's estimate $y\approx5$

$A\approx(-5,5)$

Now, let's make an educated guess about Point B.

Point E is at the origin and $AE=BE$ so the line extends the same distance from the origin in this direction as it did in the other direct.  So, we'll keep working with 5!

This $x$ is to the right of the y-axis, so it must be positive.  Let's estimate $x\approx5$

This $y$ below the x-axis, so it must be negative.  Let's estimate $y\approx-5$

$B\approx(5,-5)$

Finally, let's compare the two coordinates:

\eqalign{A&\approx(-5,5)\\B&\approx(5,-5)}

The $x$ and $y$ of each point are opposites of each other!

If $A=(x,y)$ then $B=(-x,-y)$

Overall, when dealing with coordinates and geometry, remember that everything you know about algebra (slopes, coordinates, etc.) is still true and everything you know about geometry (rectangles have right angles, lines are straight, etc.) is still true. Apply that knowledge! And don't forget to draw! It helps!