Find the distance between the points $(-3,4)$ and $(6,-2)$. (A) $3 \sqrt{5}$ (B) $3 \sqrt{13}$ (C) $\sqrt{13}$ (D) $\sqrt{85}$

Step 1 ：Given two points (-3,4) and (6,-2), we are asked to find the distance between these points.

Step 2 ：The distance between two points (x1, y1) and (x2, y2) in a plane can be calculated using the distance formula derived from the Pythagorean theorem. The formula is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). In this case, x1 = -3, y1 = 4, x2 = 6, and y2 = -2.

Step 3 ：Substitute the given values into the formula, we get \(\sqrt{(6 - (-3))^2 + ((-2) - 4)^2}\)

Step 4 ：Simplify the expression to get a decimal number, which is approximately 10.816653826391969.

Step 5 ：The options provided are in the form of square roots. Therefore, we need to convert the decimal number into a square root form to match the options. We can do this by squaring the decimal number and then simplifying the square root.

Step 6 ：The squared distance is approximately 117.00000000000001, which is not a perfect square. However, it can be expressed as 9*13, where 9 is a perfect square. Therefore, the square root of 117 can be simplified as \(3\sqrt{13}\), which matches option (B).

Step 7 ：Final Answer: \(\boxed{3 \sqrt{13}}\)