Question

Ashanti is surveying for a new parking lot shaped like a parallelogram. She knows that three of the vertices of parallelogram ABCD are A(0, 0), B(5, 2), and C(6, 5). Find the coordinates of point D and sketch parallelogram ABCD on the accompanying set of axes. Justify mathematically that the figure you have drawn is a parallelogram.

Answer 1

The sketch of the parallelogram is attached

How to show the sketch is a parallelogram

The length of opposites sides of parallelogram are equal. Also, are the slope of the lines.

The slope calculation

Slope of AB

= (2 - 0) / (5 - 0) = 2/5

Slope of CD

= (4 - 5) / (1 - 6) = 1/5

Since, slope of AB = slope of CD, AB is parallel to CD.

The lengths of AB and CD:

Length of AB

AB =
`√((5 - 0)^2 + (2 - 0)^2) = √(5^2 + 2^2) = √(25 + 4) = √(29)`

length of CD:

CD =
`CD = √((6 - 1)^2 + (5 - 4)^2) = √(5^2 + 1^2) = √(25 + 1) = √(26)`

Since AB = CD, AB is equal in length to CD.

Therefore, we have shown that opposite sides AB and CD of ABCD have the same slope and are equal in length, which means ABCD is a parallelogram.