To determine the distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a coordinate plane, we use the distance formula:

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

Given:

• Point $A(-2, 3)$
• Point $B(4, -5)$

Step 1: Substitute the values into the formula:

$d = \sqrt{(4 - (-2))^2 + (-5 - 3)^2}$

Step 2: Simplify:

$d = \sqrt{(4 + 2)^2 + (-5 - 3)^2}$ $d = \sqrt{(6)^2 + (-8)^2}$ $d = \sqrt{36 + 64}$ $d = \sqrt{100}$

Step 3: Take the square root:

$d = 10$

Conclusion:

The distance between points $A(-2, 3)$ and $B(4, -5)$ is 10 units.

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Related Questions:

1. How do you use the distance formula in 3D space?
2. What if the points are on a horizontal or vertical line?
3. How does the distance formula relate to the Pythagorean theorem?
4. Can we apply this formula to determine the distance between two points on a curve?
5. How do we calculate the midpoint between two points?

Tip: Remember, the distance formula always involves squaring the differences in both the $x$-coordinates and the $y$-coordinates, so pay attention to the signs when subtracting.