# What is the length of AB, given #A(5, -2)# and #B(-3, -4)#?

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To find the length of segment ( AB ) given the coordinates of points ( A(5, -2) ) and ( B(-3, -4) ), we can use the distance formula, which is derived from the Pythagorean theorem in a Cartesian coordinate system.

The distance formula between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is:

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

Given the coordinates of points ( A(5, -2) ) and ( B(-3, -4) ), we substitute these values into the distance formula:

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

[ d = \sqrt{(-8)^2 + (-4 + 2)^2} ]

[ d = \sqrt{64 + (-2)^2} ]

[ d = \sqrt{64 + 4} ]

[ d = \sqrt{68} ]

[ d = \sqrt{4 \times 17} ]

[ d = 2\sqrt{17} ]

Therefore, the length of segment ( AB ) is ( 2\sqrt{17} ) units.

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