Calculate Distance Using Pythagorean Theorem

In this discussion we'll learn how to find the distance between any two points in the plane given their coordinates. At the end, we'll even see how to find the distance between two points in spaaaaaace!

Let's start with a warmup:

Now do the same for a vertical line:

Notice that in the first problem, the distance is the positive difference of the $x$-coordinates, and in the second, it is the positive difference of the $y$-coordinates. Now we want a formula that works in general.

Let's try something slightly harder now.

These examples show that in general, we can compute the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ by first computing their vertical and horizontal distances, $|x_1-x_2|$ and $|y_1-y_2|$, and using the Pythagorean theorem with those as the legs!

Now, see if you can figure out the whole formula:

Bonus: The formula for the distance between two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ in three dimensions is $$\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.$$ Can you prove it?

The distance between two points: on one axis

Often, problems will require finding the distance between two points on some coordinate plane.

Let's start off with an easy case: when two points have the same $x$-coordinate or $y$-coordinate. Here, you just ignore the other axis and simply find the distance by finding the difference between the different coordinates.

Distance in $2$D

Now to tackle the harder problem: if the points had different $x$ and $y$ coordinates.

Unfortunately, the segment connecting them is not so easy to measure now. It's just too sideways.

However, there is a way around it. If we draw a right triangle with legs parallel to the $x$ and $y$ axis, there is suddenly hope. The length of the $x$-leg is simply the distance between the $x$-coordinates of the points, and similarly for the $y$-leg.

What tool do we need to proceed?

The crucial fact here is that the triangle is right. Therefore, we can use Pythagorean Theorem to figure out the length of the hypotenuse!

Which will lead us to the distance formula, but first, a problem.

The Distance Formula

Now to put all that down into one formula. Can we generalize our process to any points $(x_1, y_1)$ and $(x_2, y_2)$?

First, the $x$-leg will have a length of $|x_1-x_2|$, the difference of the $x$-coordinates. The absolute value signs are there for now since distance is always positive.

Similarly, the $y$-leg will measure $|y_1-y_2|$.

Now to use Pythagorean Theorem to get the distance. Let's call it $d$.

$$d^2=(x_1-x_2)^2+(y_1-y_2)^2$$

Note: the absolute value signs disappear because when something is squared, it is going to be positive anyway.

Finally, we square root $d^2$, giving us the distance formula:

$$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$

When most people learn this, they were usually first presented the final, complicated formula, without knowing how it works. But the explanation behind it is quite simple (and you just read one :) ).

Further Onward

Think about a $3$D coordinate space. How might we create a distance formula for two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$?

Hello! This is an example of a "portal". The idea is that once you've learned everything on this map up to here (the Distance formula), you're ready to learn the essentials of coordinate bashing.

Enjoy!

The distance formula--and hence the Pythagorean Theorem--have a neat application to geography. Here is an example.

You are traveling from Rock Springs, Wyoming to Fort Collins, Colorado. From east to west, the two cities are 290 miles apart. From north to south, the two cities are 110 miles apart. If a perfectly straight road connecting the two cities was built, how long would it be?

Solution: This problem is the distance formula in two perpendicular dimensions. They form a right triangle with legs of lengths 290 miles and 110 miles, where we seek the length of the hypotenuse, which we will label D for distance. We can use the Pythagorean Theorem to solve:

$D^2 = 290^2 + 110^2 \Rightarrow$ $D = \sqrt{290^2 + 110^2} = 310.16 \; miles$

Look at the diagram. Lets say that the coordinates of A are $(x_1,y_1)$ and the coordinates of B are $(x_2,y_2)$ Then the coordinates of C are $(x_1,y_2)$.

We will first find the length of segments AC and BC. The lengths of these segments are $y_2-y_1$ and $x_2-x_1$. Using the Pythogorean Theorem, the length of segment AB is:

$$\sqrt{(y_2-y_1)^{2}+(x_2-x_1)^{2}}$$

Q.E.D.

Here is some last-minute practice on the Distance Formula before you head to Coordinate Bashing! Best of luck on your new endeavor!