Coordinate geometry is a big focus on the ACT math section, and you’ll need to know its many facets in order to tackle the variety of coordinate geometry questions you’ll see on the test. Luckily, coordinate geometry is not difficult to visualize or wrap your head around once you know the basics. And we are here to walk you through them.
There will usually be three questions on any given ACT that involve points alone, and another two to three questions that will involve lines and slopes and/or rotations, reflections, or translations. These topics are tested by about 10% of your ACT math questions, so it is a good idea to understand the ins and outs of coordinate geometry before you tackle the test.
This article will be your complete guide to points and the building blocks for coordinate geometry: I will explain how to find and manipulate points, distances, and midpoints, and give you strategies for solving these types of questions on the ACT.
What Is Coordinate Geometry?
Geometry always takes place on a plane, which is a flat surface that goes on infinitely in all directions. The coordinate plane refers to a plane that has scales of measurement along the x and y-axes.
Coordinate geometry is the geometry that takes place in the coordinate plane.
Coordinate Scales
The x-axis is the scale that measures horizontal distance along the coordinate plane.
The y-axis is the scale that measures vertical distance along the coordinate plane.
The intersection of the two planes is called the origin.
We can find any point along the infinite span of the plane by using its position along the x and y-axes and its distance from the origin. We mark this location with coordinates, written as (x, y).
The x value tells us how far along (and in which direction) our point is along the x-axis.
The y value tells us how far along (and in which direction) our point is along the y-axis.
For instance, take look at the following graph.
This point is 4 units to the right of the origin and 2 units above the origin. This means that our point is located at coordinates (4, 2).
Anywhere to the right of the origin will have a positive x value. Anywhere left of the origin will have a negative x value.
Anywhere vertically above the origin will have a positive y value. Anywhere vertically below the origin will have a negative y value.
So, if we break up the coordinate plane into four quadrants, we can see that any point will have certain properties in terms of its positivity or negativity, depending on where it is located.
Distances and Midpoints
When given two coordinate points, you can find both the distance between them as well as the midpoint between the two original points. We can find these values by using formulas or by using other geometry techniques.
Let’s breakdown the different ways to solve these types of problems.
May you always have fast vehicles (or at least sturdy shoes) for all your distance travel.
Distance Formula
$√{(x_2-x_1)^2+(y_2-y_1)^2}$
There are two options for finding the distance between two points—using the formula, or using the Pythagorean Theorem. Let’s look at both.
Solving Method 1: Distance Formula
If you prefer to use formulas on as many questions as you are able, then go ahead and memorize the distance formula above. You will not be provided any formulas on the ACT math section, including the distance formula, so, if you choose this route, make sure you can memorize the formula accurately and call upon it as needed. (Remember—a formula you remember incorrectly is worse than not knowing a formula at all.)
You will have to memorize each and every ACT math formula you'll need and, for those of you who want to learn as few as possible, the distance formula might be the straw that broke the camel’s back. But for those of you who like formulas and have an easy time memorizing them, adding in the distance formula to your repertoire might not be a problem.
So how do we use our formula in action? Let us say we have two points, (-5, 3) and (1, -5), and we must find the distance between the two.
If we simply plug our values into our distance formula, we get:
$√{(x_2-x_1)^2+(y_2-y_1)^2}$
$√{(1-(-5))^2+(-5-3)^2}$
$√{(6)^2+(-8)^2}$
$√{(36+64)}$
$√100$
10
The distance between our two points is 10.
Solving Method 2: Pythagorean Theorem
$a^2+b^2=c^2$
Alternatively, we can always find the distance between two points by using the Pythagorean Theorem. Though, again, you won’t be given any formulas on the ACT math section, you will need to know the Pythagorean Theorem for many different types of questions, and it's a formula you’ve probably had experience using in your math classes in school. This means you will both need to know it for the test anyway, and you probably already do.
So why can we use the Pythagorean Theorem to find the distance between points? Because the distance formula is actually derived from the Pythagorean Theorem (and we'll show you how in just a bit). The trade-off is that solving your distance questions this way takes slightly longer, but it also doesn’t require you to expend energy memorizing any more formulas than you absolutely need to and carries less risk of remembering the distance formula wrong.
To use the Pythagorean Theorem to find a distance, simply turn the coordinate points and the distance between them into a right triangle, with the distance acting as a hypotenuse. From the coordinates, we can find the lengths of the legs of the triangle and use the Pythagorean Theorem to find our distance.
For example, let us use the same coordinates from earlier to find the distance between them using this method instead.
Find the distance between the points $(−5,3)$ and $(1,−5)$.
First, start by mapping out your coordinates.
Next, make the legs of your right triangles.
If we count the points along our plane, we can see that we have leg lengths of 6 and 8. Now we can plug these numbers in and use the Pythagorean Theorem to find the final piece of our triangle, the distance between our two points.
$a^2+b^2=c^2$
$6^2+8^2=c^2$
$36+64=c^2$
$100=c^2$
$c=10$
The distance between our two points is, once again, 10.
[Special Note: If you are familiar with your triangle shortcuts, you may have noticed that this triangle was what we call a 3-4-5 triangle multiplied by 2. Because it is one of the regular right triangles, you technically don’t even need the Pythagorean Theorem to know that the hypotenuse will be 10 if the two legs are 6 and 8. This is a shortcut that can be useful to know, but is not necessary to know, as you can see.]
Midpoint Formula
$(
