To find the area of the rectangle defined by the points \( A(-3, -1) \), \( B(-2, -4) \), \( C(4, -2) \), and \( D(3, 1) \), we first need to determine the lengths of the sides of the rectangle.
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Identifying vertices vs. rectangle configuration:
The points need to be labeled properly as the corners of the rectangle. By visually plotting or analyzing, we can rearrange these points:- \( A(-3, -1) \)
- \( B(-2, -4) \)
- \( C(4, -2) \)
- \( D(3, 1) \)
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Finding the lengths of the sides:
We will calculate the distances between the points to find the side lengths. The rectangle's area is given by the product of its length and width.-
Distance between points \( A \) and \( B \): \[ d_{AB} = \sqrt{(-2 - (-3))^2 + (-4 - (-1))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Distance between points \( A \) and \( D \): \[ d_{AD} = \sqrt{(3 - (-3))^2 + (1 - (-1))^2} = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]
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Area Calculation: Assuming \( d_{AB} \) represents one dimension (width), and \( d_{AD} \) represents the other dimension (length): \[ \text{Area} = \text{Width} \times \text{Length} = d_{AB} \times d_{AD} = \sqrt{10} \times 2\sqrt{10} = 2 \times 10 = 20 \]
Hence, the approximate area of the rectangle formed by the points is:
\[ \boxed{20} \]