Find the area of the polygon defined by the coordinates (4, -1), (-1, 4), (11, 16), and (16, 11).

120 square units

Step-by-step explanation:

There are a number of ways to do this. I found it convenient to let a graphing program calculate the area to be 120 square units.

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If you look at the rectangle (with sides parallel to the x- and y-axes) that encloses the figure, you see it is 17 units high and wide. The white space to the upper left and lower right together constitute a square 12 units on a side. Likewise, the white space to the upper right and lower left together comprise a square 5 units on a side. Then the shaded area is ...

17^2 - 12^2 - 5^2 = 289 -144 - 25 = 120 . . . . square units

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You can also use the Pythagorean theorem to find the length of the short side is 5√2 and the length of the long side is 12√2. Then the product of these lengths is the area:

(5√2)(12√2) = 5·12·2 = 120 . . . . square units

4 and 5)16.97 or 12√2

6)15.59 or 9√3

7)y = 15.59 or 9√3     x =27

8)17.80