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 demonstrate two things: first, it has opposite sides that are parallel, and second, its diagonals are perpendicular.

1. Parallelogram:
To prove that the opposite sides of the parallelogram are parallel, we can calculate the slopes of two pairs of opposite sides and show that they are equal.

a) Side AB:
The slope of AB can be calculated using the formula: Slope (m) = (y2 - y1) / (x2 - x1).
For points A(-5, -1) and B(-9, 6), the slope of AB is (6 - (-1)) / (-9 - (-5)) = 7 / -4.

b) Side CD:
The slope of CD can be calculated using the same formula: Slope (m) = (y2 - y1) / (x2 - x1).
For points C(-1, 5) and D(3, -2), the slope of CD is (-2 - 5) / (3 - (-1)) = -7 / 4.

The slopes of AB and CD are both 7 / -4, which means they are equal. So, we have shown that opposite sides AB and CD are parallel.

c) Side BC:
The slope of BC is (-2 - 6) / (3 - (-9)) = -8 / 12 = -2 / 3.

d) Side AD:
The slope of AD is (-1 - 5) / (-5 - 3) = -6 / -8 = 3 / 4.

The slopes of BC and AD are -2 / 3 and 3 / 4, respectively. These slopes are not equal, which indicates that sides BC and AD are not parallel.

Since the opposite sides AB and CD are parallel, but BC and AD are not parallel, we can conclude that parallelogram ABCD is not a rhombus.

Additionally, to confirm that the diagonals are perpendicular, we can calculate the slopes of the diagonals and check if they are negative reciprocals of each other. However, since the opposite sides are not parallel, we don't need to proceed with this step.