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Verify that parallelogram ABCD with vertices A(−5, −1), B(−9, 6), C(−1, 5), and D(3, −2)
is a rhombus by showing that it is a parallelogram with perpendicular diagonals.
To show that parallelogram ABCD is a rhombus, we need to show that it is both a parallelogram and has perpendicular diagonals.
1. To show that ABCD is a parallelogram, we need to show that the opposite sides are parallel. We can use the slope formula to determine if the slopes of opposite sides are equal.
- The slope of AB = (6 - (-1))/(-9 - (-5)) = 7/-4 = -7/4
- The slope of DC = (-2 - 5)/(3 - (-1)) = -7/4
- The slope of AD = (-2 - (-1))/(3 - (-5)) = -1/8
- The slope of BC = (5 - 6)/(-1 - (-9)) = 1/8
Since the slopes of both pairs of opposite sides are equal, ABCD is a parallelogram.
2. To show that ABCD has perpendicular diagonals, we need to show that the slopes of the diagonals are negative reciprocals of each other.
- The slope of AC = (5 - (-1))/(-1 - (-5)) = 6/4 = 3/2
- The slope of BD = (6 - (-2))/(-9 - 3) = 8/-12 = -2/3
Since the slopes of AC and BD are negative reciprocals of each other, the diagonals are perpendicular.
Since parallelogram ABCD has both pairs of opposite sides parallel and its diagonals perpendicular, it is a rhombus.