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To calculate the distance between the two points \( Q(-9, 9) \) and \( M(-2, 6) \), we can use the distance formula:

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

Where:
- \( (x_1, y_1) \) are the coordinates of point \( Q(-9, 9) \)
- \( (x_2, y_2) \) are the coordinates of point \( M(-2, 6) \)

Now, plug in the values:

\[
d = \sqrt{(-2 - (-9))^2 + (6 - 9)^2}
\]
\[
d = \sqrt{(-2 + 9)^2 + (-3)^2}
\]
\[
d = \sqrt{7^2 + (-3)^2}
\]
\[
d = \sqrt{49 + 9}
\]
\[
d = \sqrt{58}
\]

The exact distance between the points \( Q \) and \( M \) is \( \sqrt{58} \).

Would you like more details or have any questions? Here are some related questions to further explore this topic:

1. What are some practical applications of the distance formula in real-world problems?
2. How does the distance formula relate to the Pythagorean theorem?
3. Can you derive the distance formula from the Pythagorean theorem?
4. What would the midpoint between points \( Q \) and \( M \) be?
5. How would the distance calculation change if the points were in 3D space?

**Tip:** To find the distance between two points in a 3D space, you can extend the formula by adding a third dimension: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \).

Calculate the distance between the points Q=(-9, 9) and M=(-2, 6) in the coordinate plane. Give an exact answer (not a decimal approximation).

Learn how to calculate the exact distance between points Q=(-9, 9) and M=(-2, 6) using the distance formula. This problem involves geometry and the coordinate plane, suitable for students in Grades 9-11.

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