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.
To verify that parallelogram ABCD is a rhombus, we need to show that it is a parallelogram and that its diagonals are perpendicular to each other.
Step 1: Show that ABCD is a parallelogram.
The opposite sides of a parallelogram are parallel. We can determine the slopes of two pairs of opposite sides and show that they are equal.
The slope of side AB can be determined using the formula: slope = (y2 - y1) / (x2 - x1).
Slope of AB = (6 - (-1)) / (-9 - (-5)) = 7 / (-4) = -7/4.
The slope of side CD can be determined using the same formula:
Slope of CD = ((-2) - 5) / (3 - (-1)) = -7/4.
Since the slopes of AB and CD are equal, we can conclude that AB and CD are parallel.
Similarly, we can find the slopes of sides BC and AD to show that they are also parallel. The slopes of BC and AD are respectively -1/2 and -1/2, which are equal. Therefore, BC and AD are parallel.
Since all pairs of opposite sides are parallel, we have shown that ABCD is a parallelogram.
Step 2: Show that the diagonals of ABCD are perpendicular.
The slope of the diagonal AC can be found using the slope formula:
Slope of AC = (5 - (-1)) / (-1 - (-5)) = 6/4 = 3/2.
The slope of the diagonal BD can be found using the same formula:
Slope of BD = ((-2) - 6) / (3 - (-9)) = (-8) / 12 = -2/3.
Since the product of the slopes of the diagonals is (-2/3) * (3/2) = -1, which is the negative reciprocal of each other, we can conclude that the diagonals AC and BD are perpendicular to each other.
Therefore, by showing that ABCD is a parallelogram with perpendicular diagonals, we have verified that it is a rhombus.