To find the distance between two points $(-9, -6)$ and $(-2, -2)$ in a coordinate plane, we use the distance formula:

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

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Given:

• $(x_1, y_1) = (-9, -6)$
• $(x_2, y_2) = (-2, -2)$

Substitute these values into the distance formula:

$d = \sqrt{((-2) - (-9))^2 + ((-2) - (-6))^2}$

Simplify the expressions inside the parentheses:

$d = \sqrt{(-2 + 9)^2 + (-2 + 6)^2}$

$d = \sqrt{(7)^2 + (4)^2}$

Calculate the squares:

$d = \sqrt{49 + 16}$

Add the values:

$d = \sqrt{65}$

The distance between the points $(-9, -6)$ and $(-2, -2)$ is $\sqrt{65}$.

Approximate Value

If you need an approximate decimal value:

$d \approx 8.06$

Would you like more details, or do you have any questions?

Here are some related questions for further exploration:

1. How do you derive the distance formula?
2. How do you find the midpoint between two points?
3. What is the difference between distance and displacement in coordinate geometry?
4. How can the distance formula be extended to three-dimensional space?
5. How is the distance formula related to the Pythagorean theorem?

Tip: When working with negative coordinates, carefully follow the rules for adding and subtracting negative numbers to avoid mistakes.