This problem has a multitude of possible solutions, many of which were submitted. Here is a sampling (note that, except the last one, none of these are 'complete' solutions; they are just summaries of the methods):

Using the distance formula, discover that the sides fit a² + b² = c², so it's a right triangle. Multiply the legs, divide by two.

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Notice that two of the slopes of the sides are negative reciprocals, so the triangle is right. Ditto above.

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Use Heron's formula. This is somewhat like using a sledgehammer to kill a mosquito, but it does work.

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Draw a rectangle around triangle, and then subtract out the areas of the triangles which lie between the triangle and the rectangle.

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And finally we have this solution, which was submitted by CJ - EoB and was used (in one form or another) by several solvers:

To find the area of figures defined by coordinates such as this, list out the coordinates in this way:

 4  6
 0  9
-5 -6
 4  6

Then cross multiply on both diagonals and find the difference:

|(4*9 + 0*-6 + -5*6) - (6*0 + 9*-5 + -6*4)|
= |(36 - 30) - (-45 - 24)|
= |6- -69|
= 75

Then, divide this total by 2 to find the area:

75/2 = 37.5