Question

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

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, \(\sqrt{65} \approx 8.06\).