The distance on the coordinate plane is the length of the line connecting two points, a start point and an endpoint. This distance can be found by using a formula and substituting the coordinates of the two given points. In turn, the distance formula is derived using the Pythagorean theorem on the coordinate plane.

Here, we will learn about the formula that we can use to find the distances on the coordinate plane. Then, we will use this formula to solve some practical problems.

## How to find distances on the coordinate plane?

A distance on the coordinate plane can be found if we know the coordinates of the starting and ending points of that distance. Therefore, we can use the following steps to find distances on the coordinate plane:

**Step 1:** Determine the coordinates of the two given points on the plane. For example, we can have the points $latex A = (x_{1}, y_{1})$ and $latex B= (x_{2}, y_{2})$.

** Step 2:** Apply the distance formula to find the distance between the given points.

Distance formula$latex d=\sqrt{{{(x_{2}-x_{1})}^2}+{{(y_{2}-y_{1})}^2}}$ |

The distance formula is derived using the Pythagorean theorem, where the hypotenuse of a right triangle is equal to the distance and the legs are equal to the distances on the *x*-axis and the *y*-axis.

## Distances on the coordinate plane – Examples with answers

The following examples can be used to learn the process used to find distances on the coordinate plane. Try to solve the examples yourself before looking at the answer.

**EXAMPLE 1**

What is the distance between the points (2, 1) and (6, 4) on the coordinate plane?

##### Solution

We can write the given points in the following way:

- $latex (x_{1}, y_{1})=(2, 1)$
- $latex (x_{2}, y_{2})=(6, 4)$

Therefore, we have to use the distance formula with the given values and we have:

$latex d=\sqrt{{{(x_{2}-x_{1})}^2}+{{(y_{2}-y_{1})}^2}}$

$latex =\sqrt{{{(6-2)}^2}+{{(4-1)}^2}}$

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

$latex =\sqrt{16+9}$

$latex =\sqrt{25}$

$latex =5$

The distance between the given points is 5.

**EXAMPLE 2**

If we have the points (3, 5) and (8, 10), what is the distance on the coordinate plane?

##### Solution

Similar to the previous exercise, we can write in the following way:

- $latex (x_{1}, y_{1})=(3, 5)$
- $latex (x_{2}, y_{2})=(8, 10)$

Now, we use the coordinates of the points given in the distance formula and we have:

$latex d=\sqrt{{{(x_{2}-x_{1})}^2}+{{(y_{2}-y_{1})}^2}}$

$latex =\sqrt{{{(8-3)}^2}+{{(10-5)}^2}}$

$latex =\sqrt{{{(5)}^2}+{{(5)}^2}}$

$latex =\sqrt{25+25}$

$latex =\sqrt{50}$

$latex =7.07$

The distance is equal to 7.07.

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**EXAMPLE 3**

Determine the distance between the points (8, 4) and (6, 10).

##### Solution

In this case, we have the coordinates:

- $latex (x_{1}, y_{1})=(8, 4)$
- $latex (x_{2}, y_{2})=(6, 10)$

Using these values in the distance formula, we have:

$latex d=\sqrt{{{(x_{2}-x_{1})}^2}+{{(y_{2}-y_{1})}^2}}$

$latex =\sqrt{{{(6-8)}^2}+{{(10-4)}^2}}$

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

$latex =\sqrt{4+36}$

$latex =\sqrt{40}$

$latex =6.325$

The distance on the coordinate plane is equal to 6.325.

**EXAMPLE 4**

If we have the points (-4, -2) and (4, 7), what is their distance?

##### Solution

We can write as follows:

- $latex (x_{1}, y_{1})=(-4, -2)$
- $latex (x_{2}, y_{2})=(4, 7)$

In this case, we have a point with negative coordinates, however, we can calculate its distance in the same way using the formula:

$latex d=\sqrt{{{(x_{2}-x_{1})}^2}+{{(y_{2}-y_{1})}^2}}$

$latex =\sqrt{{{(4-(-4))}^2}+{{(7-(-2))}^2}}$

$latex =\sqrt{{{(8)}^2}+{{(9)}^2}}$

$latex =\sqrt{64+81}$

$latex =\sqrt{145}$

$latex =12.04$

The distance is equal to 12.04.

**EXAMPLE 5**

What is the distance on the coordinate plane of the points (-5, -7) and (-2, 1)?

##### Solution

We write the given coordinates as follows:

- $latex (x_{1}, y_{1})=(-5, -7)$
- $latex (x_{2}, y_{2})=(-2, 1)$

Similar to the previous exercise, we simply have to use the distance formula even if we have negative coordinates.

$latex d=\sqrt{{{(x_{2}-x_{1})}^2}+{{(y_{2}-y_{1})}^2}}$

$latex =\sqrt{{{(-2-(-5))}^2}+{{(1-(-7))}^2}}$

$latex =\sqrt{{{(3)}^2}+{{(8)}^2}}$

$latex =\sqrt{9+64}$

$latex =\sqrt{73}$

$latex =8.54$

The distance equals 8.54.

## Distance of the coordinate plane – Practice problems

Apply the distance formula to solve the following distance problems on the coordinate plane. Select an answer and check it to see if it is correct.

## See also

Interested in learning more about distance, midpoint, and slope on the plane? Take a look at these pages:

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