Question

Parallelogram ABCD has vertices: A(-3, 1), B(3, 3), C(4, 0), and D(-2, -2). In two or more complete sentences, explain how you can use the coordinates of the vertices to prove that parallelogram ABCD is a rectangle.

Answer 1

You can use these coordinates to prove that it is a rectangle because the points are pretty much all in the same space. Also, you have two negative domain's and two positive domain's.

Answer 2

Answer:

The opposite sides are equal in length and the angle between two adjacent sides is 90.

Step-by-step explanation:

Use the distance formula between two points.

Distance between A and B

AB = $\sqrt\left ( 3+3 \right )^{2}+\left ( 3-1 \right )^{2}$

AB = $\sqrt{40}$

Distance between C and D

CD = $\sqrt\left ( -2-0 \right )^{2}+\left ( 4+2 \right )^{2}$

CD = $\sqrt{40}$

Distance between C and D

BC = $\sqrt\left ( 4-3 \right )^{2}+\left ( 0-3 \right )^{2}$

BC = $\sqrt{10}$

Distance between D and A

DA = $\sqrt\left ( -2+3 \right )^{2}+\left ( -2-1 \right )^{2}$

DA = $\sqrt{10}$

Here, we observe that, AB = CD and BC = DA

Now to prove that they line AB is perpendicular to BC,

slope of AB = $\frac{3-1}{3+3}$ = \frac{1}{3}

Slope of BC = $\frac{0-3}{4-3}$ = -3

As the product of slope of AB and BC is -1 , so they are perpendicular to each other.

Now to prove that AD is perpendicular To DC

Slope of AD = $\frac{-2-1}{-2+3}$ = -3

Slope of DC = $\frac{0+2}{4+2}$ =\frac{1}{3}

As the product of slope of AD and DC is -1 , so they are perpendicular to each other.

So, it is a rectangle.