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.
AB and CD have slope -7/4
BC and AD have slope -1/8
AC has slope 3/2 and BD has slope -2/3
all four sides have length √65
To verify that parallelogram ABCD is a rhombus, we need to show that it is a parallelogram with perpendicular diagonals.
Step 1: Prove that ABCD is a parallelogram:
In order to prove that a quadrilateral is a parallelogram, we need to show that opposite sides are parallel. We can use the slope formula to find the slopes of the sides and compare them.
The slope of side AB can be calculated as:
mAB = (y2 - y1) / (x2 - x1) = (6 - (-1)) / (-9 - (-5)) = 7 / -4 = -7/4
The slope of side CD can be calculated as:
mCD = (y2 - y1) / (x2 - x1) = (-2 - 5) / (3 - (-1)) = -7 / 4 = -7/4
The slope of side BC can be calculated as:
mBC = (y2 - y1) / (x2 - x1) = (5 - 6) / (-1 - (-9)) = -1 / 8 = -1/8
The slope of side AD can be calculated as:
mAD = (y2 - y1) / (x2 - x1) = (-2 - (-1)) / (3 - (-5)) = -1 / 8 = -1/8
Since the opposite sides AB and CD have the same slope of -7/4, and the opposite sides BC and AD have the same slope of -1/8, we can conclude that ABCD is a parallelogram.
Step 2: Prove that the diagonals of ABCD are perpendicular:
To show that the diagonals of ABCD are perpendicular, we need to demonstrate that the product of the slopes of the diagonals is -1.
The slope of diagonal AC can be calculated as:
mAC = (y2 - y1) / (x2 - x1) = (5 - (-1)) / (-1 - (-5)) = 6 / 4 = 3/2
The slope of diagonal BD can be calculated as:
mBD = (y2 - y1) / (x2 - x1) = (6 - (-2)) / (-9 - 3) = 8 / (-12) = -2/3
The product of the slopes of the diagonals AC and BD is (3/2) * (-2/3) = -1.
Since the product of the slopes of the diagonals is -1, we can conclude that the diagonals of ABCD are perpendicular.
Therefore, parallelogram ABCD is a rhombus, as it has both parallel sides and perpendicular diagonals.
To verify that parallelogram ABCD is a rhombus, we need to show two things:
1. It is a parallelogram.
2. Its diagonals are perpendicular.
Let's start with the first condition.
1. To prove that ABCD is a parallelogram:
- Recall that a parallelogram is a quadrilateral with opposite sides parallel.
- We can find the slopes of AB and CD to check if they are equal.
Slope of AB:
m₁ = (y₂ - y₁) / (x₂ - x₁) = (6 - (-1)) / (-9 - (-5)) = 7 / -4 = -7/4
Slope of CD:
m₂ = (y₂ - y₁) / (x₂ - x₁) = (-2 - 5) / (3 - (-1)) = -7 / 4 = -7/4
Since the slopes are equal, AB and CD are parallel.
- Now, let's find the slopes of BC and AD.
Slope of BC:
m₃ = (y₂ - y₁) / (x₂ - x₁) = (5 - 6) / (-1 - (-9)) = -1/8
Slope of AD:
m₄ = (y₂ - y₁) / (x₂ - x₁) = (-1 - 6) / (-5 - 3) = -7/8
Again, the slopes are equal, BC and AD are also parallel.
Since opposite sides are parallel, ABCD is a parallelogram.
Now, let's move to the second condition.
2. To prove that the diagonals of ABCD are perpendicular:
- A diagonal is perpendicular to another diagonal if the product of their slopes is -1.
Slope of AC:
m₁ = (y₂ - y₁) / (x₂ - x₁) = (5 - (-1)) / (-1 - (-5)) = 6/4 = 3/2
Slope of BD:
m₂ = (y₂ - y₁) / (x₂ - x₁) = (-2 - 6) / (3 - (-9)) = -8/12 = -2/3
The product of the slopes is (3/2) * (-2/3) = -1, which means the diagonals AC and BD are perpendicular.
Since ABCD is a parallelogram with perpendicular diagonals, we have proved that it is a rhombus.