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. (3 points)
well, find the slopes of AC and BD.
If their product is -1, they are perpendicular.
To verify that parallelogram ABCD is a rhombus, we need to demonstrate that it is a parallelogram with perpendicular diagonals.
1. Proving that ABCD is a parallelogram:
- To determine if ABCD is a parallelogram, we need to show that opposite sides are parallel.
- Use the slope formula: The slope of a line passing through two points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1).
- Calculate the slopes of the lines AB, BC, CD, and DA:
Slope of AB = (6 - (-1)) / (-9 - (-5)) = 7 / (-4) = -7/4
Slope of BC = (5 - 6) / (-1 - (-9)) = -1 / 8 = -1/8
Slope of CD = (-2 - 5) / (3 - (-1)) = -7 / 4 = -7/4
Slope of DA = (-1 - (-2)) / (-5 - 3) = 1 / (-8) = -1/8
- If opposite sides have the same slope, they are parallel. Therefore, AB and CD are parallel, as well as BC and DA.
2. Proving that ABCD has perpendicular diagonals:
- To prove that the diagonals are perpendicular, we need to demonstrate that the product of the slopes of the diagonals is -1.
- Calculate the slopes of the diagonals AC and BD:
Slope of AC = (5 - (-1)) / (-1 - (-5)) = 6 / 4 = 3/2
Slope of BD = (-2 - 6) / (3 - (-9)) = -8 / 12 = -2/3
- If the product of the slopes of the diagonals is -1, then the diagonals are perpendicular. Thus, AC and BD are perpendicular.
Since ABCD is both a parallelogram and has perpendicular diagonals, we can conclude that it is a rhombus.