Find the distance between the two points in simplest radical form.
left parenthesis, minus, 9, comma, 9, right parenthesis, and , left parenthesis, 0, comma, minus, 3, right parenthesis
(−9,9) and (0,−3)

To find the distance between two points \((-9, 9)\) and \((0, -3)\), we use the **distance formula**:

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

Substitute the coordinates \((-9, 9)\) and \((0, -3)\) into the formula:

- \(x_1 = -9\), \(y_1 = 9\)
- \(x_2 = 0\), \(y_2 = -3\)

\[
d = \sqrt{(0 - (-9))^2 + (-3 - 9)^2}
\]

Simplify the expressions inside the square root:

\[
d = \sqrt{(0 + 9)^2 + (-3 - 9)^2}
\]
\[
d = \sqrt{9^2 + (-12)^2}
\]
\[
d = \sqrt{81 + 144}
\]
\[
d = \sqrt{225}
\]
\[
d = 15
\]

Thus, the distance between the two points is \( \boxed{15} \).

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. What is the distance formula derived from?
2. How would the distance change if both points were closer together?
3. Can you explain how the Pythagorean theorem relates to the distance formula?
4. What are the coordinates of a midpoint between these two points?
5. How would you calculate the distance if the points were in three-dimensional space?

**Tip**: Always double-check the signs of the coordinates to avoid simple errors when applying the distance formula.

Find the distance between the two points in simplest radical form. (−9,9) and (0,−3)

Learn how to calculate the distance between the points (-9,9) and (0,-3) using the distance formula. This step-by-step solution simplifies the problem using key geometric concepts, including the Pythagorean theorem. Suitable for students in Grades 8-10.

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