# Midpoint and Distance Formula on the Coordinate Plane

In the Cartesian Coordinate System, we take two different ponts and . These two points form a line segment with endpoints and .

We know that each line segment has the following two properties:

• Length (the distance between the endpoints)
• Midpoint (the point in the center of the line segment)

In this lesson, we will see how to calculate the length of the segment and how to find the coordinates of the midpoint, which is very easy by applying the rules that we will see shortly.

## Distance Formula

To measure the distance between two points in an orthogonal coordinate system, we use a formula that is very important in geometry. This formula is actually extracted from the Pythagorean Theorem as it is written below:

In which:

• is the length of the hypotenuse of a right triangle (the longest side).
• and the length of the two others sides (the shortest sides).

As a reminder, the Pythagorean Theorem only applies to a right-angled triangle

### Note:

Let and two points. The distance between and is denoted by , and we have:

But how did we get this formula? Let's first look at the figure below.

• The number is the length of the hypotenuse (Distance between and )
• The number is the length of the horizontal leg.
• The number is the length of the vertical leg.

So, applying the Pythagorean theorem, we have:

Remember that the distance is always positive.

## Midpoint Formula

Using geometry tools (for example, Compass), we can easily determine the midpoint of a line segment. But isn't there another way, without using geometry tools, that we can determine the midpoint?

Yes, of course, there is an arithmetic method that depends on knowing the coordinates of the two endpoints of the line segment in an orthogonal coordinate system, using a formula to calculate the coordinates of the center.

We take two points and in an orthogonal coordinate system, and take the point as the midpoint of the line segment .

### Note:

We say that is the midpoint of the line segment if the following two conditions are met together:

• There is the same distance between the two endpoints of the line segment, i.e.
• The point belongs to the line segment .

To calculate the midpoint coordinates, we use the following property:

### Note:

Let and two points and the midpoint of the line segment . We have:

The following figure shows the coordinates of the midpoint of a segment in a coordinate plane.

Let's take some examples to calculate distance and midpoint coordinates.

Example1:

Consider the points and .

1. Calculate the distance of .
2. Calculate the coordinates of the midpoint of the segment .
3. Deduce the lengths of the two segments and .

#### Solution:

1) Plotting the two points and in the coordinate plane.

Let and ; then we substitute in the distance formula:

2) We calculate the coordinates of using the midpoint formula:

3) is the midpoint of the line segment , This means that there is the same distance from both endpoints of the segment

Therefore,

Example2:

Consider the points and .

1. Calculate the distance of .
2. Calculate the coordinates of the midpoint of the segment .
3. Deduce the lengths of the two segments and .

#### Solution:

1) Pltting the two points and in the coordinate plane

.

Let and ; then we substitute in the distance formula:

2) We calculate the coordinates of using the midpoint formula:

3) is the midpoint of the line segment , This means that there is the same distance from both endpoints of the segment.

Therefore,

Example3:

Consider the points and .

1. Calculate the distance between and the origin.
2. Calculate the distance between and the origin.

#### Solution:

1) The origin of the coordinate plane is the point .

Let and ; then we substitute in the distance formula:

2) In the same way we calculate the distance between and the origin.

Let and ; then we substitute in the distance formula:

Example4:

Consider the points and .

1. Determine the coordinates of if is the midpoint of the line segment
2. Calculate the distance of the line segment

#### Solution:

1) Unlike examples 1 and 2, in this example we have the coordinates of the midpoint but we don't know the coordinates of the endpoint of the line segment . To solve this problem, it's very easy, we just apply the Midpoint Formula but this time differently.

We have and let

Since is the midpoint of then:

The coordinates on both sides of the equation are equal. It means that the -coordinate on the left side is equal to the -coordinate on the right side.

The mathematical expression is:

And the -coordinate on the left side is equal to the -coordinate on the right side.

The mathematical expression is:

Here we get two simple equations:

• The unknown in equation (1) is
• The unknown in equation (2) is

Let's solve equation (1), we have

Let's solve equation (2), we have:

Finally, the coordinates of the endpoint of the line segment are

2) Let's calculate the distance of the line segment

Let and

Therefore, the length of the line segment is: