Let and be the coordinates of a line segment . We have:

Simple, right? Very simple. Straight to examples.

#### Example 1

Find the midpoint of the line segment joining and .

#### Example 2

Find the midpoint of and .

#### Example 3

Find the midpoint of the line segment joining and

#### Example 4

Find the midpoint of the line segment joining and

#### Example 5

Find the midpoint of the line segment joining and

#### Example 6

A diameter of a circle has endpoints located at and . Find the coordinates of the center of the circle.

The center of a circle is at the __midpoint__ of any of its diameters, so we simply have to find the midpoint of and in this case; it is:

So this circle is centered at , the origin.

#### Example 7

How many line segments have midpoints at the origin ?

The answer is __infinite__. In fact, for any number , the points and all have midpoints at

which is the origin. In general, this also works for any point other than the origin: once the midpoint is specified, there are infinite line segments that share that midpoint. On the other hand, a given line segment will always have a unique midpoint.

In the terminology of functions (don’t worry about this yet if you’re still in grade 10), the assignment

is a function, while its inverse is not.

#### Example 8

Line segment has midpoint at . If is the point , find the coordinates of .

Let be the point with coordinates . Since the midpoint of and is , we have, by the midpoint formula, that

The -coordinates must be equal, so ; similarly, the -coordinates must be equal, so .

Therefore, is the point .

#### Example 9

Prove that the midpoint of and lies on the line .

We first calculate the midpoint of the given points and ; it is:

Next, we check whether the midpoint we obtained, namely , is on the given line . To do this, let’s substitute the value of (that is, ) into the given equation and see if we’ll get the value of (that is, ):

Since we obtained the desired value of , we can conclude that the midpoint lies on the line .

#### Example 10

If the midpoint of and is , determine the values of and .

We have, by the midpoint formula, that:

This in turn leads to two separate equations:

Clear fractions to obtain

If we re-arrange a bit, we see that this is a linear system:

Adding the two equations: . Substituting in any of the two equations gives . So we conclude that and .

## Exercises for the reader

- Derive the midpoint formula; that is, prove that the midpoint of and is .
- An idea from statistics: Let be the midpoint of and . Define the lower midpoint to be the midpoint of and , and the upper midpoint to be the midpoint of and . Prove that:
- In furtherance of exercise 2 above, prove that the midpoint of and is , the midpoint of and . (This is expected.)