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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 verify that parallelogram ABCD with vertices A (-5, -1), B (-9, 6), C (-1, 5), and D (3, -2) is a rhombus, we need to show that it is a parallelogram and that its diagonals are perpendicular.
1. To show that ABCD is a parallelogram, we need to show that opposite sides are parallel.
- The slope of AB = (6 - (-1))/(-9 - (-5)) = 7/(-4) = -7/4
- The slope of CD = (-2 - 5)/(3 - (-1)) = -7/4
- Since the slopes are equal, AB and CD are parallel.
- The slope of BC = (5 - 6)/(-1 - (-9)) = 1/8
- The slope of AD = (-2 - (-1))/(3 - (-5)) = -1/8
- Since the slopes are equal, BC and AD are parallel.
Therefore, opposite sides AB and CD are parallel, and opposite sides BC and AD are parallel. Thus, ABCD is a parallelogram.
2. To show that the diagonals AC and BD are perpendicular, we need to show that their slopes are negative reciprocals of each other.
- The slope of AC = (5 - (-1))/(-1 - (-5)) = 6/4 = 3/2
- The slope of BD = ((-2) - 6)/(3 - (-9)) = (-8)/12 = -2/3
- The product of the slopes is (3/2) * (-2/3) = -1
Since the product is -1, the diagonals AC and BD are perpendicular.
Therefore, we have shown that ABCD is a parallelogram with perpendicular diagonals, which means it is a rhombus.