Chapter 3: Graphing

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 (a^2 + b^2 = c^2) to find the distance.

To do this for the two data points (x_1, y_1) and (x_2, y_2), the distance between these two points (d) will be found using \Delta x = x_2 - x_1 and \Delta y = y_2 - y_1.

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

d^2 = \Delta x^2 + \Delta y^2

image

or, in expanded form:

d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2

 

image

On graph paper, this looks like the following. For this illustration, both \Delta x and \Delta y are 7 units long, making the distance d^2 = 7^2 + 7^2 or d^2 = 98.

image

The square root of 98 is approximately 9.899 units long.

Example 3.2.1

Find the distance between the points (-6,-4) and (6, 5).

Start by identifying which are the two data points (x_1, y_1) and (x_2, y_2). Let (x_1, y_1) be (-6,-4) and (x_2, y_2) be (6, 5).

Now:

\Delta x^2 = (x_2 - x_1)^2 or [6 - (-6)]^2 and \Delta y^2 = (y_2 - y_1)^2 or [5 - (-4)]^2.

This means that

d^2 = [6 - (-6)]^2 + [5 - (-4)]^2

or

d^2 = [12]^2 + [9]^2

which reduces to

d^2 = 144 + 81

or

d^2 = 225

Taking the square root, the result is d = 15.

Finding the Midway Between Two Points (Midpoint)

The logic used to find the midpoint between two data points (x_1, y_1) and (x_2, y_2) on a graph involves finding the average values of the x data points (x_1, x_2) and the of the y data points (y_1, y_2). The averages are found by adding both data points together and dividing them by 2.

In an equation, this looks like:

x_{\text{mid}}=\dfrac{x_2+x_1}{2} and y_{\text{mid}}=\dfrac{y_2+y_1}{2}

Example 3.2.2

Find the midpoint between the points (-2, 3) and (6, 9).

image

We start by adding the two x data points (x_1 + x_2) and then dividing this result by 2.

x_{\text{mid}} = \dfrac{(-2 + 6)}{2}

or

\dfrac{4}{2} = 2

The midpoint’s y-coordinate is found by adding the two y data points (y_1 + y_2) and then dividing this result by 2.

y_{\text{mid}} = \dfrac{(9 + 3)}{2}

or

\dfrac{12}{2} = 6

The midpoint between the points (-2, 3) and (6, 9) is at the data point (2, 6).

Questions

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

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

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

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

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-3-2/”>Answer Key 3.2

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