Question 1131976: The coordinates of the vertices of parallelogram ABCD are A(-1,-3), B(7,2), C(5,8), D(-3,3). Using the diagonals, determine whether ABCD is a rhombus. Show all your work and state appropriate formulas and theorems used.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Note even a very rough sketch of the four given points, without doing any calculations, suggests that the figure might be a parallelogram or even a rectangle, but not a rhombus.
For ABCD to be a rhombus, we must have all three of the following:
(1) AB parallel to CD (slopes the same)
(2) BC parallel to AD (slopes the same)
(3) AC perpendicular to BD (product of slopes is -1)
If any ONE of these is not satisfied, the figure is not a rhombus.
(1) slope of AB: 5/8; slope of CD: 5/8 -- okay
(2) slope of BC: 6/2 = 3; slope of AD: 6/2 = 3 -- okay
(3) slope of AC: 11/6; slope of BD: -1/10 -- NOT okay
The figure is indeed a parallelogram, because opposite pairs of sides are parallel. But the slopes of the diagonals are not perpendicular.
ANSWER: The figure is not a rhombus, because the diagonals are not perpendicular.
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