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To find the distance between the two points \( P_1(3, -3) \) and \( P_2(2, 3) \), we can use the distance formula, which is:

\[
d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Here, the coordinates of \( P_1 \) are \( (x_1, y_1) = (3, -3) \) and the coordinates of \( P_2 \) are \( (x_2, y_2) = (2, 3) \).

### Step-by-Step Solution:
1. Subtract the x-coordinates: \( x_2 - x_1 = 2 - 3 = -1 \)
2. Subtract the y-coordinates: \( y_2 - y_1 = 3 - (-3) = 3 + 3 = 6 \)
3. Square both differences: 
   - \( (-1)^2 = 1 \)
   - \( 6^2 = 36 \)
4. Add the squared values: \( 1 + 36 = 37 \)
5. Take the square root: \( \sqrt{37} \)

Thus, the exact distance is:

\[
d(P_1, P_2) = \sqrt{37}
\]

Would you like more details on this? 

Here are 5 related questions to expand your understanding:
1. How can you find the midpoint between two points?
2. How do you derive the distance formula from the Pythagorean theorem?
3. What is the geometric interpretation of \( \sqrt{37} \) as a distance in 2D space?
4. How can you calculate distances in 3D space between points?
5. What are examples of other real-world uses of the distance formula?

**Tip:** Always double-check your signs when working with coordinates to avoid errors!

Find the distance between the points P1 and P2. P1 = (3, -3); P2 = (2, 3).

Learn how to calculate the exact distance between points P1(3, -3) and P2(2, 3) using the distance formula derived from the Pythagorean theorem. Step-by-step explanation provided. Suitable for students in Grades 9-10.

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