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

Distance Formula
The distance from point A to point B is found using the distance formula and by using Pythagorean Theorem.

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.