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:

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

or, in expanded form:

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

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 \(d^2 = 98\).

The square root of 98 is approximately 9.899 units long.

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 \(y_{\text{mid}}=\dfrac{y_2+y_1}{2}\)

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.

9. (−6, −1) and (6, 5)
10. (1, −4) and (5, −2)
11. (−5, −1) and (3, 5)
12. (6, −4) and (12, 4)
13. (−8, −1) and (6, 7)
14. (1, −6) and (3, −2)
15. (−7, −1) and (3, 9)
16. (2, −2) and (12, 4)

Answer Key 3.2