To compute the area of the rectangle defined by the coordinates \( A(-3,-1) \), \( B(-2,-4) \), \( C(4,-2) \), and \( D(3,1) \), we first need to confirm that the given points indeed form a rectangle.
Next, we can identify the length and width of the rectangle based on the coordinates. The area of a rectangle can be computed using the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Step 1: Identify Points
Given points are:
- \( A = (-3, -1) \)
- \( B = (-2, -4) \)
- \( C = (4, -2) \)
- \( D = (3, 1) \)
Step 2: Find Lengths of the Sides
We need to find the lengths of the sides of the shape formed by these points. The exact pairing of the vertices is critical, but we will assume these points need to form a typical rectangle configuration.
Find distances between the points:
- Distance \( AB \): \[ AB = \sqrt{((-2) - (-3))^2 + ((-4) - (-1))^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
- Distance \( BC \): \[ BC = \sqrt{(4 - (-2))^2 + ((-2) - (-4))^2} = \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]
- Distance \( CD \): \[ CD = \sqrt{(3 - 4)^2 + (1 - (-2))^2} = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
- Distance \( DA \): \[ DA = \sqrt{((-3) - 3)^2 + ((-1) - 1)^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]
Step 3: Area Calculation
From the calculations above, we can observe:
- Opposite sides \( AB \) and \( CD \) are both \( \sqrt{10} \).
- Opposite sides \( BC \) and \( DA \) are both \( 2\sqrt{10} \).
Since these dimensions correspond to a rectangle:
- Length = \( 2\sqrt{10} \)
- Width = \( \sqrt{10} \)
Now, calculate the area: \[ \text{Area} = \text{Length} \times \text{Width} = (2\sqrt{10}) \times (\sqrt{10}) = 2 \times 10 = 20 \]
Conclusion
The approximate area of the rectangle defined by the points \( A \), \( B \), \( C \), and \( D \) is \( \boxed{20} \).
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