Answer:
(d) 34 square units
Explanation:
There are several ways to determine the correct choice of answer. You can estimate the area, or make use of the fact that the vertices are on grid points. You can also figure the area by finding the lengths of the sides of the rectangle and using the area formula for a rectangle.
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estimate
Each long side has a vertical extent of about 8 units. The left side, for example, extends from y=-4 to y=+4. Each short side has a horizontal extent of 4 units. The top extends from x=-1 to x=3, for example. The product of these lengths is an approximation of the area of the rectangle:
8×4 = 32
This tells you the area is not any of the first three answer choices:
The fact that each side length is slightly longer than the estimated value we used means the area is more than 32. 34 square units is the best estimate.
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Pick's theorem
When a figure is bounded by straight lines and has vertices on grid points, the area can be found exactly using Pick's theorem. It tells you the area is ...
A = i +b/2 -1
where i is the number of interior grid points and b is the number of grid points on the boundary.
The boundary lines cross the grid at an angle, so only intercept certain grid points. The four vertices are on grid points (of course), and there is a grid point on the boundary at the midpoint of each side. That's a total of 6 boundary grid points.
One can count the interior grid points, or recognize there is a 4×4 array of them on and above the x-axis, and a similar array below the x-axis. That's a total of 32 interior grid points.
By Pick's theorem, the area of the rectangle is ...
A = i +b/2 -1 = 32 +6/2 -1 = 34 . . . square units
Note that Pick's theorem is telling you any figure with vertices on the grid will always have an area that is a whole number of half square units. The area will never be irrational.
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using side lengths
As we noted above, the horizontal extent of the top and bottom sides is 4 units. Each of those sides has a corresponding vertical extent of 1 unit. Its length is the hypotenuse of a right triangle with legs 4 and 1, so can be found using the Pythagorean theorem:
c² = a² +b²
c = √(a² +b²) = √(4² +1²) = √17
Looking carefully at the long sides of the rectangle, we see that each of those is double this length. Then the rectangle's area is ...
A = bh = (√17)(2√17) = 2·17 = 34 . . . square units