Verify that parallelogram abcd with vertices a ( − 5 , − 1 ) , b ( − 9 , 6 ) , c ( − 1 , 5 ) , and d ( 3 , − 2 ) 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.

Step-by-step explanation:

The diagonals of the parallelogram are A(-5, -1), C(-1, 5) and B(-9, 6), D(3, -2).

Slope of diagonal AC = (5 - (-1)) / (-1 - (-5)) = (5 + 1) / (-1 + 5) = 6 / 4 = 3/2

Slope of diagonal BD = (-2 - 6) / (3 - (-9)) = -8 / (3 + 9) = -8 / 12 = -2/3

For the parallelogram to be a rhombus, the intersection of the diagonals are perpendicular.

i.e. the product of the two slopes equals to -1.

Slope AC x slope BD = 3/2 x -2/3 = -1.

Therefore, the parallelogram is a rhombus.

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