Problem

Find the distance between the two points. Simplify your answer, and write the exact answer in simplest radical form for irrational answers.
\[
(x, y)=(-8,3) \text { and }(x, y)=(3,-9)
\]

Answer

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Answer

Final Answer: The distance between the two points is \(\boxed{\sqrt{265}}\)

Steps

Step 1 :Given two points in a 2D plane, (-8, 3) and (3, -9), we are asked to find the distance between these points.

Step 2 :The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula derived from the Pythagorean theorem: \(d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\)

Step 3 :Substitute the given points into the formula: \(d = \sqrt{(3 - (-8))^2 + ((-9) - 3)^2}\)

Step 4 :Simplify the expression inside the square root: \(d = \sqrt{(11)^2 + (-12)^2}\)

Step 5 :Calculate the squares: \(d = \sqrt{121 + 144}\)

Step 6 :Add the results: \(d = \sqrt{265}\)

Step 7 :Final Answer: The distance between the two points is \(\boxed{\sqrt{265}}\)

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