Explanation:
To create a rectangle with vertices located at (-5, 3), (-5, -7), and (5, -7), we need to find the fourth vertex that completes the rectangle. Since opposite sides of a rectangle are parallel and congruent, we can determine the missing vertex by finding the midpoint of either of the two given sides and then moving in the direction perpendicular to that side by the length of the other side.
The given sides are:
Side 1: (-5, 3) to (-5, -7), which has length 3 - (-7) = 10
Side 2: (-5, -7) to (5, -7), which has length 5 - (-5) = 10
Since the sides are congruent, we can find the midpoint of Side 1 as:
Midpoint of Side 1 = [(-5 + (-5))/2, (3 + (-7))/2] = [-5, -2]
To find the missing vertex, we need to move from (-5, -2) in the direction perpendicular to Side 1 by a distance of 10 units (the length of Side 2). Since Side 2 is horizontal, we need to move vertically. We can do this by adding or subtracting 10 from the y-coordinate of the midpoint of Side 1, depending on whether we want to move up or down. In this case, we want to move down, so we subtract 10:
Missing vertex = [-5, -2 - 10] = [-5, -12]
Therefore, the fourth vertex needed to create a rectangle with vertices located at (-5, 3), (-5, -7), and (5, -7) is (-5, -12).