Distance and Midpoints

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 $(x,y)$.

A graph showing the points (2, 1) and (4, 3) - StudySmarter Originals

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:

A line segment connecting the points (2, 1) and (4, 3) - StudySmarter Originals

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 of a line segment connecting the points (2, 1) and (4, 3) - StudySmarter Originals

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 ($6÷2=3$). This gives us the x coordinate of the midpoint, 3. Then, we add the two y coordinates ($1+3=4$) and divide by two ($4÷2=2$) 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.

A graph showing how Pythagoras' Theorem can be used to calculate the distance between two points in the coordinate plane - StudySmarter Originals

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.

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.