Find the distance between the points (-5, 2) and (-6, 10).

Question

You can use the distance formula, but it is not on the formula sheet and is hard to use.

It is easier to plot it on the coordinate plane - Use the Desmos graphing calculator to plot the points, drop the right triangle, and use the Pythagorean Theorem.

Answer 1

To find the distance between the points $(-5,2)$ and $(-6,10)$, you can use the distance formula, but we'll go through the visual method you mentioned as well.

1. Plotting the Points:

   • Point A: $(-5,2)$
   • Point B: $(-6,10)$

2. Dropping the Right Triangle:

   • From point A, draw a vertical line down to the y-coordinate of point B (which is 10). The horizontal distance between the two points corresponds to the change in the x-coordinates, while the vertical distance corresponds to the change in the y-coordinates.

3. Calculating Distances:

   • The change in x (horizontal distance) = $|-6-(-5)|=|-6+5|=1$
   • The change in y (vertical distance) = $|10-2|=8$

4. Using the Pythagorean Theorem:

   • Let $a=1$ (horizontal distance) and $b=8$ (vertical distance). Then the distance $d$ can be calculated using the Pythagorean Theorem: $$d=\sqrt{a^2+b^2}=\sqrt{1^2+8^2}=\sqrt{1+64}=\sqrt{65}$$

Thus, the distance between the points $(-5,2)$ and $(-6,10)$ is $\sqrt{65}$.

If you need a numerical approximation, $\sqrt{65}\approx 8.06$.