# 3.2 Midpoint and Distance Between Points

# Finding the Distance Between Two Points

The logic used to find the distance between two data points on a graph involves the construction of a right triangle using the two data points and the Pythagorean theorem to find the distance.

To do this for the two data points and , the distance between these two points will be found using and

Using the Pythagorean theorem, this will end up looking like:

or, in expanded form:

On graph paper, this looks like the following. For this illustration, both and are 7 units long, making the distance or .

The square root of 98 is approximately 9.899 units long.

Example 3.2.1

Find the distance between the points and .

Start by identifying which are the two data points and . Let be and be .

Now:

or and or .

This means that

or

which reduces to

or

Taking the square root, the result is .

# Finding the Midway Between Two Points (Midpoint)

The logic used to find the midpoint between two data points and on a graph involves finding the average values of the data points and the of the data points . The averages are found by adding both data points together and dividing them by .

In an equation, this looks like:

and

Example 3.2.2

Find the midpoint between the points and .

We start by adding the two data points and then dividing this result by 2.

or

The midpoint’s -coordinate is found by adding the two data points and then dividing this result by 2.

or

The midpoint between the points and is at the data point .

# Questions

For questions 1 to 8, find the distance between the points.

- (−6, −1) and (6, 4)
- (1, −4) and (5, −1)
- (−5, −1) and (3, 5)
- (6, −4) and (12, 4)
- (−8, −2) and (4, 3)
- (3, −2) and (7, 1)
- (−10, −6) and (−2, 0)
- (8, −2) and (14, 6)

For questions 9 to 16, find the midpoint between the points.

- (−6, −1) and (6, 5)
- (1, −4) and (5, −2)
- (−5, −1) and (3, 5)
- (6, −4) and (12, 4)
- (−8, −1) and (6, 7)
- (1, −6) and (3, −2)
- (−7, −1) and (3, 9)
- (2, −2) and (12, 4)