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 is a rhombus, we need to show that it is a parallelogram and that its diagonals are perpendicular.
First, we can find the slopes of the opposite sides of the parallelogram to show that it is a parallelogram.
The slope of side AB = (6 - (-1)) / (-9 - (-5)) = 7 / (-4) = -7/4
The slope of side CD = (-2 - 5) / (3 - (-1)) = -7 / 4
The slope of side BC = (5 - 6) / (-1 - (-9)) = -1 / 8
The slope of side AD = (-1 - (-2)) / (-5 - 3) = 1 / 8
Since the slopes of opposite sides are equal, we can conclude that ABCD is a parallelogram.
Next, we can find the slopes of the diagonals to show that they are perpendicular.
The slope of diagonal AC = (5 - (-1)) / (-1 - (-5)) = 6 / 4 = 3/2
The slope of diagonal BD = (-2 - 6) / (3 - (-9)) = -8 / 12 = -2/3
Since the product of the slopes of the diagonals is (3/2) * (-2/3) = -1, we can conclude that the diagonals are perpendicular.
Therefore, parallelogram ABCD is a rhombus.