Step-by-step explanation:
To find the area of the quadrilateral \( ABCD \), we can split it into two triangles and then use the formula for the area of a triangle.
Let's call the coordinates of the points \( A(-3, 2) \), \( B(4, 2) \), \( C(3, -3) \), and \( D(-3, 3) \).
First, we find the length of \( AB \) and \( BC \) using the distance formula:
\[ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]
\[ = \sqrt{(4 - (-3))^2 + (2 - 2)^2} \]
\[ = \sqrt{(7)^2 + (0)^2} \]
\[ = \sqrt{49} \]
\[ = 7 \]
\[ BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} \]
\[ = \sqrt{(3 - 4)^2 + (-3 - 2)^2} \]
\[ = \sqrt{(-1)^2 + (-5)^2} \]
\[ = \sqrt{1 + 25} \]
\[ = \sqrt{26} \]
Now, we can find the area of triangle \( ABC \) using the formula for the area of a triangle:
\[ Area_{ABC} = \frac{1}{2} \times AB \times BC \]
\[ = \frac{1}{2} \times 7 \times \sqrt{26} \]
Similarly, we find the length of \( CD \) and \( DA \):
\[ CD = \sqrt{(x_D - x_C)^2 + (y_D - y_C)^2} \]
\[ = \sqrt{(-3 - 3)^2 + (3 - (-3))^2} \]
\[ = \sqrt{(-6)^2 + (6)^2} \]
\[ = \sqrt{36 + 36} \]
\[ = \sqrt{72} \]
\[ DA = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2} \]
\[ = \sqrt{(-3 - (-3))^2 + (2 - 3)^2} \]
\[ = \sqrt{(0)^2 + (-1)^2} \]
\[ = \sqrt{1} \]
\[ = 1 \]
Now, we can find the area of triangle \( ACD \):
\[ Area_{ACD} = \frac{1}{2} \times CD \times DA \]
\[ = \frac{1}{2} \times \sqrt{72} \times 1 \]
Finally, to find the area of quadrilateral \( ABCD \), we sum the areas of triangles \( ABC \) and \( ACD \):
\[ Area_{ABCD} = Area_{ABC} + Area_{ACD} \]
\[ Area_{ABCD} = \frac{1}{2} \times 7 \times \sqrt{26} + \frac{1}{2} \times \sqrt{72} \]
\[ Area_{ABCD} = \frac{7\sqrt{26}}{2} + \frac{\sqrt{72}}{2} \]
\[ Area_{ABCD} = \frac{7\sqrt{26} + \sqrt{72}}{2} \]
\[ Area_{ABCD} \approx \frac{7 \times 5.099 + 8.485}{2} \]
\[ Area_{ABCD} \approx \frac{35.693 + 8.485}{2} \]
\[ Area_{ABCD} \approx \frac{44.178}{2} \]
\[ Area_{ABCD} \approx 22.089 \]
Therefore, the area of the quadrilateral \( ABCD \) is approximately \( 22.089 \) square units.
