To determine the distance between the two points on the graph, we will use the distance formula, which is:

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

Step 1: Identify the coordinates of the two points.

From the graph:

• The first point $P_1$ has coordinates $(-5, -3)$,
• The second point $P_2$ has coordinates $(2, 1)$.

Step 2: Apply the distance formula.

Substitute $(x_1, y_1) = (-5, -3)$ and $(x_2, y_2) = (2, 1)$ into the formula:

$d = \sqrt{(2 - (-5))^2 + (1 - (-3))^2}$

Simplify the differences:

$d = \sqrt{(2 + 5)^2 + (1 + 3)^2}$

$d = \sqrt{7^2 + 4^2}$

Square the values:

$d = \sqrt{49 + 16}$

$d = \sqrt{65}$

Step 3: Simplify the radical.

The square root of $65$ cannot be simplified further because $65$ factors as $5 \times 13$, both of which are primes.

Thus, the distance between the two points is:

$\sqrt{65}$

Step 4: Match with the answer choices.

None of the given options include $\sqrt{65}$ directly. It's possible that either the choices or the points may have been misinterpreted. Could you confirm if you'd like me to revisit this graph or recheck the question? Let me know!

Related Questions:

1. How do you simplify square roots involving prime factorizations?
2. What is the distance formula and why is it derived from the Pythagorean Theorem?
3. How can you graphically estimate distances on a coordinate plane?
4. What is the significance of radical expressions in geometry?
5. How can you calculate midpoints between two points?

Tip: Always verify the coordinates carefully when applying the distance formula. Each error can compound the result.