To solve the problem of finding the distance between two points, we will use the distance formula, which is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

9) Points: (-3, 6) and (-8, 3)

Using the formula: $d = \sqrt{((-8) - (-3))^2 + (3 - 6)^2}$ $d = \sqrt{(-8 + 3)^2 + (3 - 6)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}$ So, the distance is: $d = \sqrt{34} \approx 5.83$

10) Points: (4, 0) and (8, 4)

Using the formula: $d = \sqrt{((8) - (4))^2 + (4 - 0)^2}$ $d = \sqrt{(8 - 4)^2 + (4 - 0)^2} = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32}$ So, the distance is: $d = \sqrt{32} \approx 5.66$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the distance formula derived from the Pythagorean theorem?
2. Can you find the midpoint between the same pairs of points?
3. What is the significance of distance in geometry, particularly in coordinate systems?
4. How can you calculate the distance between two points in three-dimensional space?
5. How would the distance formula change if one of the coordinates were variables?

Tip: To avoid calculation mistakes, always ensure you subtract the coordinates in the same order and carefully square the results.