To find the area of the rectangle given by the vertices \((0, 0)\), \((2, 2)\), \((9, -4)\), and \((7, -6)\), we first need to check if these points actually create a rectangle and then determine the lengths of the sides.
Step 1: Determine the lengths of the sides
The points can be grouped into pairs of opposite corners of the rectangle. To see if the opposite points \((0, 0)\) and \((9, -4)\), and \((2, 2)\) and \((7, -6)\) form a rectangle, we compute the distances of the sides formed by these points.
Distance between \((0, 0)\) and \((2, 2)\):
\[ d = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
Distance between \((0, 0)\) and \((7, -6)\):
\[ d = \sqrt{(7 - 0)^2 + (-6 - 0)^2} = \sqrt{49 + 36} = \sqrt{85} \]
Distance between \((2, 2)\) and \((9, -4)\):
\[ d = \sqrt{(9 - 2)^2 + (-4 - 2)^2} = \sqrt{49 + 36} = \sqrt{85} \]
Distance between \((2, 2)\) and \((7, -6)\):
\[ d = \sqrt{(7 - 2)^2 + (-6 - 2)^2} = \sqrt{25 + 64} = \sqrt{89} \]
Step 2: Area calculation
Since we expect a rectangle, we can calculate the area by multiplying one pair of opposite sides.
The length of the sides calculated turns out to be different than expected; we will check if the area stays consistent.
Area calculation with coordinates
Using the formula for the area \(A\) of a polygon defined by vertices \((x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4)\) \[ A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \] where \( (0,0) \), \( (2,2) \), \( (9,-4) \), \( (7,-6) \):
\[ A = \frac{1}{2} \left| 02 + 2(-4) + 9*(-6) + 70 - (02 + 2*9 + (-4)*7 + (-6)*0) \right| \] Calculating the first part: \[ 0 + (-8) + (-54) + 0 = -62 \]
Calculating the second part: \[ 0 + 18 + 28 + 0 = 46 \]
Thus, the Area becomes: \[ A = \frac{1}{2} \left| -62 - 46 \right| = \frac{1}{2} \left| -108 \right| = \frac{108}{2} = 54 \]
However, with needs for verifying bounds, let’s see the expected area from lists:
Each option \(A. \ 52.2 \ B. 25.76 \ C. 24.1 \ D. 12.05\)
By double-checking lengths or possibly known cases of rectilinear areas further can help. Yet it appears our estimate for significant areas runs comparably around bounds. The area from dimensions reflects best is around \( 52.2\).
Thus, the suitable question lookups or checks can consistently prompt output closer to:
Final Conclusion: Approximate area is best represented near: A. 52.2 units^2.
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