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## Distance and midpoints: Definition

A **line segment** is a part of a line which joins two different points together. It is only a segment of a line, as lines are infinitely long.

The **midpoint** of a line segment that connects two points is the middle point that lies on the line segment.

A line segment AB with midpoint C - StudySmarter Originals

The above line segment AB connects the points A and B. Point C lies on the middle point of the line, so it is the midpoint of the line.

The **distance**** between two points** is the length of the line segment that joins the points together.

A line segment AB - StudySmarter Originals

The distance between point A and point B above is the length of the blue line, which is the line segment that connects the two points.

## Distance and midpoints in the coordinate plane

In the coordinate plane, points are defined by an $x$ coordinate and a $y$ coordinate which indicate how far across and how far up the coordinate plane the line is located, respectively. Coordinate points are written in the form $(x,y)$.

Shown in the figure above are the coordinate points $(2,1)$ and $\left(4,3\right)$. Let's draw a line segment between them in the following figure:

The middle point on this line segment is the midpoint, and the length of the line segment describes the distance between the two points. How can we determine the middle point on the line and the distance between the two points? There are formulas that can help us work these details out. Let's take a look at them.

## Distance and midpoints: Formula

In this section we look at two formulas:

- The formula for the midpoint of a line segment
- The formula for the distance between two points

### Formula for the midpoint of a line segment

We know that the midpoint is located halfway between the two points. This means that its x coordinate is halfway between the x coordinates of the points, and its y coordinate is halfway between the y coordinates of the points. So, if we know the x coordinates and y coordinates for both points, how can we find the midpoint that is located halfway between them?

For a line segment connecting points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$, the midpoint is: $({x}_{m},{y}_{m})=(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2})$.

The figure below shows points $(2,1)$ and $\left(4,3\right)$ on a coordinate plane, with a midpoint at $(3,2)$. Let's discuss in more detail how the midpoint is found.

The midpoint formula given above allows us to find the **average** of the points' x coordinates and the **average** of the points' y coordinates. In order to find the average for our points' x coordinates, we must add the two x coordinate values. ($2+4=6$) and then divide by two ($6\xf72=3$). This gives us the x coordinate of the midpoint, 3. Then, we add the two y coordinates ($1+3=4$) and divide by two ($4\xf72=2$) to find the average of our points' y coordinates, which is the midpoint's y coordinate.

### Formula for the distance between two points

When we work out the distance between two points, we are finding the length of the line segment between them. How do we determine this length?

The distance between two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is given by the formula: $r=\sqrt{{({x}_{2}-{x}_{1})}^{2}+({y}_{2}-{y}_{1}}{)}^{2}$.

Therefore, if we know the x and y coordinates of both points, we can apply this formula. You may be wondering where this formula comes from. The formula for the distance between points considers the points' connecting line segment as if it were a hypotenuse of a right triangle, which means that the Pythagoras' theorem applies, given by ${a}^{2}+{b}^{2}={c}^{2}$. The hypotenuse a and its accompanying side lengths b and c can be seen in the figure below.

The formula also considers the difference between the points' x coordinates and y coordinates, which can be obtained by subtracting the smaller coordinates from the larger ones in this case. In the graph above, the distance between the x coordinates of $\left(2,1\right)$ and $\left(4,3\right)$ is $4-2=2$. This means that our imaginary triangle has a side length of 2! Similarly, the distance between the y coordinates is $3-1=2$, which gives a second triangle side length of 2.

What about the triangle's hypotenuse? When we apply the Pythagoras' theorem, we acquire the length of the hypotenuse of the triangle we have created, which will be the line segment between the two points:

$\sqrt{{2}^{2}+{2}^{2}}=\sqrt{8}=2\sqrt{2}$

Now that we understand the origins of the formula, let's apply it in an example problem.

## Finding midpoints and distance with examples

Using the previously provided formulas, we can find distance and midpoints between two points defined in the coordinate plane.

**Find the midpoint of the line segment that joins the points $(3,6)$ and $\left(8,-4\right)$**

**Solution:**

If we know the coordinates of two points, we can use the midpoint formula to work out the midpoint of the line segment connecting them. The midpoint formula is given by $({x}_{m},{y}_{m})=(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2})$, so we substitute our known coordinates into the formula to get the midpoint.

For the x coordinate of the midpoint, $\left({x}_{m}\right)=\left(\frac{{x}_{1}+{x}_{2}}{2}\right)$ =$\frac{3+8}{2}=\frac{11}{2}$. Here, we have added the values of the x coordinates and divided by two.

For the y coordinate of the midpoint,$\left({y}_{m}\right)=\left(\frac{{y}_{1}+{y}_{2}}{2}\right)=\frac{6+(-4)}{2}=\frac{2}{2}=1$. Here, we have added the values of the y coordinates of each of the points. Remember, adding a negative number is the same as subtraction, and any number over itself can be simplified to one.

Now that we have our x and y coordinate of the midpoint, we can write it in the typical form (x, y).

$(\frac{11}{2},1)$

**Find the exact distance between the points $\left(-2,5\right)$ and $\left(7,8\right)$**

**Solution:**

We can apply the distance formula to find the distance between two points. The distance formula is given by $r=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$. We substitute our coordinates into the formula to get the distance. It doesn't matter which order we use to subtract the coordinates from each other (which point is considered 1 or 2). What matters is that we stay consistent in subtracting an x coordinate from the other x coordinate, and in subtracting a y coordinate from the other y coordinate. This is because when we square the values it gets rid of the negative sign and ends up as the same value each way.

$\sqrt{(7-{(-2))}^{2}+{(8-5)}^{2}}=\sqrt{81+9}=\sqrt{90}=3\sqrt{10}$ Here we have substituted the x and y coordinates of each point into the distance formula to get the distance between them. The question asked for the exact distance, so we leave our result as an irrational number.

## Applications of the midpoint and distance formula

Midpoints and distances have many mathematical applications. In this section we provide some examples of how the formulas we discussed can be applied.

### Using the midpoint formula to finding the perpendicular bisector

You can use the midpoint formula to find the perpendicular bisector of a line segment. A perpendicular bisector is a line that intersects with a line segment at its midpoint at a right angle, cutting it into two halves.

We can find a perpendicular bisector by finding a line segment's midpoint and slope as well as the negative reciprocal of its slope. Then we substitute these values into the formula $y-{y}_{1}=m(x-{x}_{2})$, using the coordinates of the midpoint and the negative reciprocal of the line segment's slope.

### Using the distance formula to define shapes and objects

We know that we can use the distance formula to find the lengths of line segments if we know the points where the line segment starts and ends. For example, if we had a ladder that was resting against a wall, and the ladder touched the wall and floor at points $(A,\hspace{0.17em}B)$ and $\left(C,D\right)$ we could use the distance formula to determine the ladder's length.

We can also prove that the line segments joining points on a coordinate plane work together to form a specific shape. For example, if three line segments joining 3 points all have equal distances, we can show that the line segments form an equilateral triangle, as equilateral triangles have 3 equal side lengths.

**A ladder is resting against a wall in the xy plane, with the x axis representing the floor and the y axis representing a wall. It is touching the floor at the point $(0,3)$ and is touching the wall at the point $\left(0,5\right)$. Find the length of the ladder to 2 decimal places. **

Solution:

These two points represent the ends of the ladder as they are both touching either the floor or the wall. To find the length of the ladder we need to find the distance between the two points. We do this by using the distance formula.

$\sqrt{{(0-3)}^{2}+{(5-0)}^{2}}=\sqrt{9+25}=\sqrt{34}$

We have substituted our points into the distance formula to get the distance between them, which represents the length of the ladder. The question indicates that we should give our answer to 2 decimal places, so we convert the irrational number to decimal form, giving us 5.83. The length of the ladder is 5.83 to 2 decimal places.

## Distance and midpoints - Key takeaways

- A line segment is part of a line. It joins two points together.
- The midpoint of a line segment is the middle point that lies on it.
- The distance between two points is the length of the line segment that connects the two points together.
- The midpoint of the line segment connecting the points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is given by the formula $({x}_{m},{y}_{m})=(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2})$
- The distance between the points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ is given by the formula $r=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}$

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##### Frequently Asked Questions about Distance and Midpoints

What is the distance and midpoint formula?

The distance between two points is the square root of the sum of the difference of the x coordinates squared and the difference of the y coordinates squared.

The midpoint of the line segment between two points has an x coordinate of the sum of the x coordinates of the points divided by two, and a y coordinate of the sum of the y coordinates of the points divided by two.

How to find midpoint and distance in geometry?

You can use the midpoint and distance formulas to find midpoint and distance, as long as you know the coordinates of two points.

The midpoint formula is:.

The distance formula is: .

What are some examples of distance and midpoints?

An example of distance is the distance between two boats located on the surface of the sea.

An example of a midpoint is the center of a circle, as it occurs at the midpoint of the circle's diameter.

How to find distance and midpoint between two points?

You can use the midpoint and distance formulas to find midpoint and distance, as long as you know the coordinates of two points.

What is distance and midpoint?

The distance between two points is the length of the line segment connecting them.

The midpoint of a line segment is the middle point that lies on the line segment.

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