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To find the distance between two points \( P_1 = (5, -3) \) and \( P_2 = (3, 3) \), we use the distance formula:

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

Substituting in the coordinates:

- \( x_1 = 5 \), \( y_1 = -3 \)
- \( x_2 = 3 \), \( y_2 = 3 \)

\[
d(P_1, P_2) = \sqrt{(3 - 5)^2 + (3 - (-3))^2}
\]

First, calculate each part:

\[
(3 - 5)^2 = (-2)^2 = 4
\]
\[
(3 - (-3))^2 = (3 + 3)^2 = 6^2 = 36
\]

Now, add these together:

\[
4 + 36 = 40
\]

Take the square root:

\[
d(P_1, P_2) = \sqrt{40}
\]

We can simplify \(\sqrt{40}\):

\[
\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}
\]

Thus, the distance is:

\[
d(P_1, P_2) = 2\sqrt{10}
\]

Would you like any more details or explanations on how this calculation works?

Here are 5 related questions to expand your understanding:
1. How do you derive the distance formula between two points in a coordinate plane?
2. Can you use the distance formula in three dimensions? How does it change?
3. How can you determine if three points are collinear using distances?
4. What are some real-life applications of the distance formula?
5. How is the midpoint between two points calculated?

**Tip:** When working with the distance formula, always ensure to carefully subtract and square each component, as it helps prevent sign errors that might impact the result.

Find the distance d(P1, P2) between the points P1 = (5, -3) and P2 = (3, 3). Simplify your answer. Type an exact answer using radicals as needed.

Learn how to calculate the distance between two points P1 = (5, -3) and P2 = (3, 3) using the distance formula. This problem covers key concepts in coordinate geometry and involves simplifying the square root to an exact answer in terms of radicals. Suitable for students in Grades 8-10.

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