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.

## 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: $d^2=3^2+4^2$ $d^2=9+16$ $d^2=25$ $\sqrt{d^2}=\sqrt{25}$ $d=5$

## 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.