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To find the distance between two points \( H(-6, 9) \) and \( N(-2, 1) \) on the coordinate plane, 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 \( H \),
- \( (x_2, y_2) \) are the coordinates of point \( N \).

### Given:
\( H = (-6, 9) \) and \( N = (-2, 1) \).

Substitute the values:

\[
d = \sqrt{((-2) - (-6))^2 + (1 - 9)^2}
\]
\[
d = \sqrt{((-2 + 6)^2 + (-8)^2)}
\]
\[
d = \sqrt{(4^2 + (-8)^2)}
\]
\[
d = \sqrt{(16 + 64)}
\]
\[
d = \sqrt{80}
\]
\[
d = 4\sqrt{5}
\]

Thus, the exact distance between points \( H \) and \( N \) is \( 4\sqrt{5} \).

Would you like further explanation on how this formula works or any other questions?

### Related Questions:
1. What is the derivation of the distance formula?
2. How does the distance formula relate to the Pythagorean theorem?
3. How do you find the midpoint between two points in the coordinate plane?
4. How would the calculation change if we were finding the distance in 3D space?
5. How can you apply the distance formula to solve real-world problems?

### Tip:
Always double-check your subtraction signs in the distance formula to avoid errors when dealing with negative coordinates!

Calculate the distance between the points H = (-6, 9) and N = (-2, 1) in the coordinate plane. Give an exact answer (not a decimal approximation).

Learn how to calculate the exact distance between two points, H(-6, 9) and N(-2, 1), using the distance formula. This problem applies key geometry concepts such as coordinate plane distance and the Pythagorean theorem. Suitable for students in Grades 8-10.

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