In this video, we are going to look at the distance formula. The distance formula is used to find the distance between two points, so in this case, the distance from A to B. The video lesson above will cover a midpoint formula example and show how the midpoint formula is related to Pythagorean Theorem. After you finish this lesson, view all of our Pre-Algebra lessons and practice problems.

## What is the Distance Formula?

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

For example:
If point A was (1,2) and point B was (4,6), then we have to find the distance between the two points.
$d=\sqrt{(4-1)^2+(6-2)^2}$
$d=\sqrt{(3)^2+(4)^2}$
$d=\sqrt{9+16}$
$d=\sqrt{25}$
$d=5$

If you don’t want to memorize the formula, then there is another way to find the distance between the two points. Draw a line from the lower point parallel to the x-axis, and a line from the higher point parallel to the y-axis, then a right triangle will be formed. Then, we can use the Pythagorean theorem to solve for the distance.

The distance of one leg is the difference in the x’s. 4-1=3 so that side of the triangle is 3. The distance of the other leg is the difference in the y’s. 6-2=4 so that side of the triangle is 4. Now we can use the Pythagorean theorem. The distance formula is actually based off of that:
$d=\sqrt{a^2+b^2}$

If we solved using the Pythagorean theorem, then:
$d^2=3^2+4^2$
$d^2=9+16$
$d^2=25$
$\sqrt{d^2}=\sqrt{25}$
$d=5$

## Examples of Distance formula

### Example 1

Find the distance between the two points $(-3,2)$ and $(3,5)$

Distance formula is:

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

Substitute the given then evaluate

$d = \sqrt{(3 - -3)^2 + (5 - 2)^2}$

$d = \sqrt{(6)^2 + (3)^2}$

$d = \sqrt{36 + 9}$

$d = \sqrt{45}$

$d=6.71$

Now, we have

$d = 6.71$

### Example 2

What is the distance between the points (4, 3) and (-4,-3)?

Distance formula is:

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

Substitute the given then evaluate

$d = \sqrt{(-4 - 4)^2 + (-3 - 3)^2}$

$d = \sqrt{(-8)^2 + (-6)^2}$

$d = \sqrt{64+36}$

$d = \sqrt{100}$

$d=10$

Now, we have

$d = 10$

## Video-Lesson Transcript

In this lesson, let’s discuss the distance formula.

The distance formula is used to find the distance between two points.

In this case, distance from point $A$ to point $B$.

We have coordinates $(x_1, y_1)$ for point $A$.

And point $B$ has coordinates $(x_2, y_2)$.

Distance formula is

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

You can memorize this formula and find the distance between any two points.

Let’s have an example.

Point $A$ has coordinates $(1, 2)$. Point $B$ has coordinates $(4, 6)$. Let’s find the distance between these two points.

Distance formula is

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$d = \sqrt{(4 - 1)^2 + (6 - 2)^2}$
$d = \sqrt{(3)^2 + (4)^2}$
$d = \sqrt{9 + 16}$
$d = \sqrt{25}$

Now, we have

$d = 5$

This means that the distance between point $A$ and point $B$ is $5$.

If you don’t want to memorize the formula, there’s another way to find the distance between the two points.

It’s very similar to the formula.

Let’s draw a triangle using the line from $A$ to point $B$.

Make sure to make to form a triangle with a $90^{\circ}$ angle.

Then, we can use the Pythagorean theorem.

The distance of the horizontal leg is the difference of the two $x$-coordinates. Let’s call it $a$.

$a = 4 - 1$ $a = 3$

Then, the distance of the vertical leg is the difference of the two $y$-coordinates. Let’s call this leg $b$.

$b = 6 - 2$
$b = 4$

Now, take a look at the Pythagorean theorem.

$a^2 + b^2 = c^2$

The distance formula is actually based on this.

So, let’s solve this using Pythagorean theorem

$d^2 = 3^2 + 4^2$
$d^2 = 9 + 16$
$d^2 = 25$

Then, let’s get the squareroot of $d$ to get the answer.

$\sqrt{d^2} = \sqrt{25}$
$d = 5$

So, we got the same answer as the first one.

So now, you could memorize the distance formula or you could draw a triangle and use the Pythagorean theorem.