Have you used the Pythagorean Theorem before? If not, make sure to check out the Pythagorean Theorem lesson. If you have, you already know how to find the distance between two points! We usually think of the Pythagorean Theorem only for a right triangle that's drawn, but you can also use it to find the distance between two points on a plane.
Let's say you have to find the distance between the two points shown in the graph below. 
If we use the grid lines to draw in lines down and across, we can form a right triangle! 
Now that we have a right triangle, we can use the Pythagorean Theorem to find the distance between A and B. The distance we want to find is the hypotenuse of the right triangle and we can just count the spaces to find the lengths of the legs. 
Let's look at another example. Let's say you have to find the distance between the points (1,2) and (6,4). First, let's plot the points on a graph and draw in the line segment between the two points. We need to find the length of this segment. 
Next, we can use the grid lines to create a right triangle. The distance we want to find is the hypotenuse of the right triangle and we can just count the spaces down and over to find the lengths of the legs. We can plug these values into the Pythagorean Theorem and solve for x. 
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The Distance Formula
We can always use a graph and the Pythagorean Theorem to find the distance between two points. But what if your coordinates are really large? Or really small decimals? It could be a big pain to try and graph the coordinates. It's also time consuming to always have to graph the coordinates first and then draw in the right triangle. Thankfully, there's another way! The Distance Formula is essentially a shortcut so you don't have to get out graph paper to find the distance between two points.
The Distance Formula is given below. Don't let it scare you! We'll look at each piece of the formula so you can see what's really going on.
The Distance Formula is given below. Don't let it scare you! We'll look at each piece of the formula so you can see what's really going on.
If we squared both sides of the distance formula, we would get this:
Does this look familiar at all? It's really just the Pythagorean Theorem in a fancy form. When you subtract the two x-values, you're really just finding the length of the horizontal leg in the right triangle. When you subtract the two y-values, you're finding the length of the vertical leg in the right triangle. (Depending on the order you subtract them in, you might get a negative value but it doesn't matter since you're going to be squaring it anyways.)
To use the formula, you subtract the x-values from the two points. This gives you your "a" value. Then subtract the two y-values from the two points. This gives you your "b" value. Don't worry if one or both are negative, you're going to square them so you'll always end up with a positive number in the end. Do a-squared plus b-squared next and take the square root to find "c" - this will be the distance between the two points.
Let's look at an example. We'll do it with a graph so you can see what's happening in each step.
Example 1
Find the distance between the points (1,0) and (4,5).
Let's start by looking at the points on a graph. We can draw a line segment between them and label it d. We need to find the length of this segment.
Let's start by looking at the points on a graph. We can draw a line segment between them and label it d. We need to find the length of this segment.
To use the Distance Formula, it can help if you label the points. It doesn't matter which point you call point 1 or point 2. It's usually easiest to just pick the first one listed to call Point 1, but you'll get the same answer either way. Remember that the x-coordinate is always listed first and the y-coordinate is listed second in an ordered pair.
Next, plug the values into the Distance Formula. You subtract the x-values (this gives you the length of the horizontal leg). Then subtract the y-values (this gives you the length of the vertical leg). Depending on which point you call Point 1 or Point 2, you will sometimes end up with negative values in this step (technically the length of the legs is the absolute value of what you get when you subtract the coordinates). You're going to square these values in the next step so you'll always end up with positive values under the square root.
Next you need to simplify what's under the square root. Square the values, add them together, then then take the square root last. You'll want to check with your teacher or look at the instructions to see if you should give the answer in simplified radical form or as a rounded decimal.
Example 2
Find the distance between the points (-3,4) and (1,-2).
Here's a graph of the points so you can visualize it. We need to find the length of the segment that connects the two points. Before you start plugging values into the Distance Formula, it helps to try and picture the right triangle that could be drawn and think about the lengths of a, b, and c if you were using the Pythagorean Theorem.
Here's a graph of the points so you can visualize it. We need to find the length of the segment that connects the two points. Before you start plugging values into the Distance Formula, it helps to try and picture the right triangle that could be drawn and think about the lengths of a, b, and c if you were using the Pythagorean Theorem.
It's usually helpful to label the points before you plug values into the formula. Remember that it doesn't matter which one you call Point 1 and which one you call Point 2. You're going to get the same answer either way. When you label the points, make sure to remember that the x-coordinate is listed first in an ordered pair.
Next, plug these values into the Distance Formula and simplify to find the distance between the two points.
Make sure to remember that subtracting is the same thing as adding the opposite. 1 - (-3) is the same as 1 + 3. Keep in mind that if the coordinates have different signs, it means they're on the opposite sides of the axis. This will help you visualize how far apart they really are. 1 and -3 are 4 units apart on the graph.
When you subtract the y-coordinates you get -6. If you look at the graph, you can see that the length of the vertical leg is a positive 6. It just depends on the order that you labeled the points, sometimes you're going to get a negative value when you subtract. The length of the leg is actually the absolute value of the difference. It doesn't matter in the end because you square the value so it will come out positive either way. If you end up with a negative value under the square root in the very end, it's a sign you did something wrong! Make sure to use parentheses if you're plugging this into a calculator.
Double check with your teacher to see if you should be giving your answer in simplified radical form (How do you simplify a radical?) or as a rounded decimal (How do you round a decimal?).