To find the area of square DEFG, we first need to verify whether the points D, E, F, and G actually form a square, and then calculate the area.
Step 1: Calculate the lengths of the sides.
The distances between the points can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Distance DE:
- \(D = (7, -5)\)
- \(E = (2, -5)\)
\[ DE = \sqrt{(2 - 7)^2 + (-5 - (-5))^2} = \sqrt{(-5)^2 + 0^2} = \sqrt{25} = 5 \]
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Distance EF:
- \(E = (2, -5)\)
- \(F = (0, 2)\)
\[ EF = \sqrt{(0 - 2)^2 + (2 - (-5))^2} = \sqrt{(-2)^2 + (7)^2} = \sqrt{4 + 49} = \sqrt{53} \]
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Distance FG:
- \(F = (0, 2)\)
- \(G = (0, 7)\)
\[ FG = \sqrt{(0 - 0)^2 + (7 - 2)^2} = \sqrt{0 + 25} = 5 \]
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Distance GD:
- \(G = (0, 7)\)
- \(D = (7, -5)\)
\[ GD = \sqrt{(7 - 0)^2 + (-5 - 7)^2} = \sqrt{(7)^2 + (-12)^2} = \sqrt{49 + 144} = \sqrt{193} \]
Step 2: Verify the square properties.
For DEFG to be a square, all sides must be equal, and the diagonals must also be equal. From the calculated distances:
- \(DE = 5\)
- \(EF = \sqrt{53}\)
- \(FG = 5\)
- \(GD = \sqrt{193}\)
Since not all sides are equal, DEFG is not a square. In fact, we cannot define the area of square DEFG as it does not exist.
Conclusion: Square DEFG does not exist with the provided points since the lengths of the sides are not equal. Therefore, we cannot calculate the area of a non-existent square.