7. 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.

(y2-y1)/(x2-x1)

that is the formula

easy,

after making sketch, show that the slopes of opposite sides are equal.
Then draw the diagonals and find their slopes
show that the slope of one diagonal is the opposite reciprocal of the slope of the other.
Be careful in your slope calculations with the subtraction of all those negative numbers

I will do one slope
slope AB = (6-(-1))/(-9 - (-5)) = 7/-4 or -7/4

what she needs is the formula...(that is helping not doing the work)

Thanks CTree for the formula

To verify that a parallelogram ABCD is a rhombus, we need to show that it is both a parallelogram and has perpendicular diagonals.

1. Checking if it is a parallelogram:
- Given the coordinates of the vertices, we can find the slopes of the opposite sides of the parallelogram.
- 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 of AB and CD are equal, AB is parallel to CD.
- The slope of BC = (5 - 6)/(-1 - (-9)) = 1/8.
- The slope of AD = (-2 - (-1))/(3 - (-5)) = -1/8.
Since the slopes of BC and AD are equal, BC is parallel to AD.
Hence, AB || CD and BC || AD.
Since both pairs of opposite sides are parallel, ABCD is a parallelogram.

2. Checking if it has perpendicular diagonals:
- The midpoint of BD can be found by taking the average of the x-coordinates and the average of the y-coordinates.
- Midpoint of BD = ([(3 - 9)/2], [(-2 + 6)/2]) = (-3, 2).
- The slope of AC = (5 - (-1))/(-1 - (-5)) = 6/4 = 3/2.
To check if the diagonals are perpendicular, we need to check if the product of their slopes is -1.
- The slope of the line passing through the midpoint of BD and A = (2 - (-1))/(-3 - (-5)) = 3/2.
The product of the slopes of AC and the line passing through the midpoint of BD and A is (3/2) * (3/2) = 9/4.
Since the product is not -1, the diagonals are not perpendicular.

Therefore, despite being a parallelogram, ABCD does not have perpendicular diagonals, which means it is not a rhombus.