What is the distance formula? A variation of the Pythagorean Theorem

The distance formula is used to find the distance between any two points on the coordinate plane. The distance formula is a variation of the Pythagorean Theorem.

What is the Distance Formula?

The distance formula is used to find the distance between any two points on the coordinate plane.

The Distance Formula

The distance between two points

x and y distance points

is given by

distance formula equation

Does the distance formula look familiar to you at all?It should be, because the distance formula is actually a variation of the Pythagorean Theorem! Recall, that the Pythagorean Theorem is

Pythagorean Theorem Equation

and can be used to find a missing side of a right triangle.

How does the Distance Formula work?

To show how the Pythagorean Theorem can be transformed into the distance formula, we first want to take our right triangle onto the coordinate plane.

Right Triangle - Pythagorean Theorem

Next, we will change our side lengths to represent distance instead. For example, the length of A in the image above is 3 units long on the x-axis. We can represent this in distance form by taking the largest x-value of the triangle and subtracting it from the smallest x-value of the triangle. The largest value (X2) and the smallest value (X1) is 0. Therefore, the distance of the length of A can be written as (3-0) or (X2-X1). This can be also done using the length of B, but the general distance will be (Y2-Y1).

Pythagorean Theorem formula 2

Finally, we can use the Pythagorean theorem to find the distance of the triangle above.

Distance Formula Problem Example 1:

Find the distance between the points (-27, 4) and (19,-6).

Step 1: Label your points.

Distance Formula - Label your points

Step 2: Substitute the values in the formula and simplify.

Distance Formula - Substitute Values and Simplify formula

Thus, the distance between (-27, 4) and (19, -6) is47.07 units long.

Distance Formula Problem Example 2:

Find the distance between the points

Step 1: Label your points.

Substitute Variables in Distance Formula

Step 2: Substitute the values in the formula and simplify.

Thus, the distance between (20,3) and (15,5) is 5.38 units long.

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