Best Answer:
Got vectors? That helps with a couple of things. Let A,B,C,D be the displacement vectors to the vertices in the order you listed them:

A = (1, 5), B = (4, 4), C = (9, -7) and D = (12, -8)

The displacements from point A are:

B-A = (3, -1)

C-A = (8, -12)

D-A = (11, -13)

You need one of those to be the sum of the other two for the given points to be the vertices of a parallelogram, and indeed (D-A) is the sum of (B-A) + (C-A). That's parallelogram addition of vectors; so AB and AC are sides of the parallelogram and AD is a diagonal.

The magnitude of the cross product (B-A)x(C-A) is the area. so compute:

(3, -1)x(8, -12) = (3)(-12) - (-1)(8) = -36 + 8 = -28

So, the parallelogram area is 28 square units.

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