# Lesson Explainer: Midpoints and Perpendicular Bisectors Mathematics

In this explainer, we will learn how to find the perpendicular bisector of a line segment by identifying its midpoint and finding the perpendicular line passing through that point.

Our first objective is to learn how to calculate the coordinates of the midpoint of a line segment connecting two points. Suppose we are given two points and . The midpoint of the line segment is the point lying on exactly halfway between and . This means that the -coordinate of lies halfway between and and may therefore be calculated by averaging the two points, giving us . The same holds true for the -coordinate of . This leads us to the following formula.

### Formula: The Coordinates of a Midpoint

Suppose and are points joined by a line segment . Then, the coordinates of the midpoint of the line segment are given by

Let us practice finding the coordinates of midpoints.

### Example 1: Finding the Midpoint of a Line Segment given the Endpoints

Given and , what are the coordinates of the midpoint of ?

We recall that the midpoint of a line segment is the point halfway between the endpoints, which we can find by averaging the - and -coordinates of and respectively. Thus, we apply the formula:

Therefore, the coordinates of the midpoint of are .

We can also use the formula for the coordinates of a midpoint to calculate one of the endpoints of a line segment given its other endpoint and the coordinates of the midpoint.

### Example 2: Finding an Endpoint of a Line Segment given the Midpoint and the Other Endpoint

The origin is the midpoint of the straight segment . Find the coordinates of point if the coordinates of point are .

Here, we have been given one endpoint of a line segment and the midpoint and have been asked to find the other endpoint. We can do this by using the midpoint formula in reverse:

This gives us two equations: and

We conclude that the coordinates of are .

One application of calculating the midpoints of line segments is calculating the coordinates of centers of circles given their diameters for the simple reason that the center of a circle is the midpoint of any of its diameters.

In the next example, we will see an example of finding the center of a circle with this method.

### Example 3: Finding the Center of a Circle given the Endpoints of a Diameter

Points and define the diameter of a circle with center . Find the coordinates of and the circumference of the circle, rounding your answer to the nearest tenth.

The center of the circle is the midpoint of its diameter . Recall that the midpoint of a line segment (such as a diameter) can be found by averaging the - and -coordinates of the endpoints and as follows:

The circumference of a circle is given by the formula , where is the length of its radius. The length of the radius is the distance from the center of the circle to any point on its radius, for example, the point . We can calculate this length using the formula for the distance between two points and :

Taking the square roots, we find that and therefore the circumference is to the nearest tenth.

In conclusion, the coordinates of the center are and the circumference is 31.4 to the nearest tenth.

We turn now to the second major topic of this explainer, calculating the equation of the perpendicular bisector of a given line segment.

### Definition: Perpendicular Bisectors

Given a line segment , the perpendicular bisector of is the unique line perpendicular to passing through the midpoint of .

Recall that for any line with slope , the slope of any line perpendicular to it is the negative reciprocal of , that is, . We can use this fact and our understanding of the midpoints of line segments to write down the equation of the perpendicular bisector of any line segment.

### How to: Calculating the Equation of the Perpendicular Bisector of a Line Segment

Suppose we are given a line segment with endpoints and and want to find the equation of its perpendicular bisector.

1. First, we calculate the slope of the line segment. To do this, we recall the definition of the slope:
2. Next, we calculate the slope of the perpendicular bisector as the negative reciprocal of the slope of the line segment:
3. Next, we find the coordinates of the midpoint of by applying the formula to the endpoints:
4. We can now substitute these coordinates and the slope into the point–slope form of the equation of a straight line:
This gives us an equation for the perpendicular bisector.

Let us have a go at applying this algorithm.

### Example 4: Finding the Perpendicular Bisector of a Line Segment Joining Two Points

Find the equation of the perpendicular bisector of the line segment joining points and . Give your answer in the form .

To find the equation of the perpendicular bisector, we will first need to find its slope, which is the negative reciprocal of the slope of the line segment joining and . This is given by

Now, we can find the negative reciprocal by flipping over the fraction and taking the negative; this gives us the following:

Next, we need the coordinates of a point on the perpendicular bisector. Since the perpendicular bisector (by definition) passes through the midpoint of the line segment , we can use the formula for the coordinates of the midpoint:

Substituting these coordinates and our slope into the point–slope form of the equation of a straight line, and rearranging into the form , we have .

For our last example, we will use our understanding of midpoints and perpendicular bisectors to calculate some unknown values.

### Example 5: Determining the Unknown Variables That Describe a Perpendicular Bisector of a Line Segment

A line segment joins the points and . The perpendicular bisector of has equation . Find the values of and .

Since the perpendicular bisector has slope , we know that the line segment has slope (the negative reciprocal of ). We can calculate the -coordinate of point (that is, ) by using the definition of the slope :

We will calculate the value of in the equation of the perpendicular bisector using the coordinates of the midpoint of (which is a point that lies on the perpendicular bisector by definition).

We have the formula

We can now substitute and into the equation of the perpendicular bisector and rearrange to find :

Our solution to the example is , .

Let us finish by recapping a few important concepts from this explainer.

### Key Points

• We can use the formula to find the coordinates of the midpoint of a line segment given the coordinates of its endpoints. We can use the same formula to calculate coordinates of an endpoint given the midpoint and the other endpoint.
• We can calculate the centers of circles given the endpoints of their diameters.
• We know that the perpendicular bisector of a line segment is the unique line perpendicular to the segment passing through its midpoint.
• We have a procedure for calculating the equation of the perpendicular bisector of a line segment given the coordinates of its endpoints:
1. We first calculate its slope as the negative reciprocal of the slope of the line segment.
2. We then find the coordinates of the midpoint of the line segment, which lies on the bisector by definition.
3. Finally, we substitute these coordinates and the slope into the point–slope form of the equation of a straight line, which gives us an equation for the perpendicular bisector.