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Three of the four vertices of a rectangle in the xy

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Three of the four vertices of a rectangle in the xy

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C

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Source: Manhattan Prep

Three of the four vertices of a rectangle in the xy-coordinate plane are (-5,1), (-4,4), and (8,0). What is the fourth vertex?

A. (-4.5, 2.5)
B. (-4, 5)
C. (6, -2)
D. (7, -3)
E. (10, 1)

The OA is D.

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Please refer to the attached figure.



Since the figure is rectangle. The lines BD and DE are perpendicular to each other (Point E is the unknown vertex)

Product of slopes of 2 lines perpendicular to each other is -1
from the figure we have slope of BD = (0 - 4)/(8-(-4)) = -4/12 = -1/3
Hence, Slope of line DE will be = (-1)/(-1/3) = 3
From the figure, we can see that x-coordinate of unknown vertex is positive while y-coordinate is negative.
Hence, A, B and E are out

Plugging in C and D to check with which vertex we can obtain slope of 3:
C : (-2-0)/(6-8) = -2/-2 = 1
D: (-3-0)(7-8) = 3

Hence, D is the correct answer

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