Quadrilateral ABCD has the following vertices: A(0,6)
B(3,5)
C(0, -4)
D(-3,-3)
Also, angle a is a right angle.
Is quadrilateral ABCD a rectangle, and why?

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  yes

Step-by-step explanation:

The figure can be shown to be a parallelogram by showing the sum of endpoints of the diagonals is the same.

  A +C = B +D

  (0, 6) +(0, -4) = (0, 2) = (3, 5) +(-3, -3) . . . . diagonals bisect each other

If the diagonals of a quadrilateral bisect each other, it is a parallelogram. A parallelogram with a right angle is a rectangle. So, ABCD is a rectangle.

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Additional comment

The midpoint of each diagonal is half the sum of the end point coordinates. That is, the midpoints are (0, 2)/2 = (0, 1). Since calculation of the midpoints requires both sums be divided by 2, we can tell the midpoints are the same if the sums are the same.

Step-by-step explanation:

a = 12 in

b = 8 in.

h = 6 in.

plugin values into the formula:

A = 1/2 h(a + b)

A = (12 + 8)  x  6

           2

A = 60 in²