To determine the area of the rectangle defined by the given points, we first need to understand whether the points form the vertices of a rectangle. The four points provided are:
- Point A: (-3, -1)
- Point B: (1, 5)
- Point C: (4, 3)
- Point D: (0, -3)
We can start by plotting these points and assessing the pairs of opposite sides to see if they are perpendicular (which is a property of rectangles) and if the distances are equal for the opposite pairs.
Calculate the distances between opposite pairs:
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Distance between A and B: \[ d_{AB} = \sqrt{(1 - (-3))^2 + (5 - (-1))^2} = \sqrt{(1 + 3)^2 + (5 + 1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \quad \text{or approximately } 7.21 \]
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Distance between C and D: \[ d_{CD} = \sqrt{(0 - 4)^2 + (-3 - 3)^2} = \sqrt{(0 - 4)^2 + (-3 - 3)^2} = \sqrt{(-4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \quad \text{or approximately } 7.21 \]
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Distance between A and D: \[ d_{AD} = \sqrt{(0 - (-3))^2 + (-3 - (-1))^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \quad \text{or approximately } 3.61 \]
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Distance between B and C: \[ d_{BC} = \sqrt{(4 - 1)^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \quad \text{or approximately } 3.61 \]
Since the lengths of opposite sides (AB and CD, AD and BC) are equal and both pairs are also perpendicular to each other, we can conclude that points A, B, C, and D do form a rectangle.
Calculate the area of the rectangle:
Area is given by: \[ \text{Area} = \text{length} \times \text{width} \]
Where:
- Length (based on distance between A and B) = \( \sqrt{52} \)
- Width (based on distance between A and D or B and C) = \( \sqrt{13} \)
So, the area is: \[ \text{Area} = \sqrt{52} \times \sqrt{13} = \sqrt{52 \times 13} = \sqrt{676} = 26 \text{ units}^2 \]
Thus, the area of the rectangle is \( \boxed{26} \) units².
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