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 $\mathrm{\Delta }x=x_2-x_1$ and $\mathrm{\Delta }y=y_2-y_1.$

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

$d^2=\mathrm{\Delta }{x}^2+\mathrm{\Delta }{y}^2$

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 $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ 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:

$x_{\text{mid}}={\displaystyle \frac{x_2+x_1}{2}}$ and $y_{\text{mid}}={\displaystyle \frac{y_2+y_1}{2}}$

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