To find the area of the rectangle formed by the points A(-3, -1), B(1, 5), C(4, 3), and D(0, -3), we first need to identify which points are the vertices of the rectangle.
Let's plot the points:
- A(-3, -1)
- B(1, 5)
- C(4, 3)
- D(0, -3)
To see which points form the rectangle, we need to check the distances between the points to determine the dimensions.
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Finding the distance between points:
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Distance AB: \[ \sqrt{(1 - (-3))^2 + (5 - (-1))^2} = \sqrt{(1 + 3)^2 + (5 + 1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \]
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Distance AC: \[ \sqrt{(4 - (-3))^2 + (3 - (-1))^2} = \sqrt{(4 + 3)^2 + (3 + 1)^2} = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65} \]
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Distance AD: \[ \sqrt{(0 - (-3))^2 + (-3 - (-1))^2} = \sqrt{(0 + 3)^2 + (-3 + 1)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
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Distance BC: \[ \sqrt{(4 - 1)^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
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Distance BD: \[ \sqrt{(0 - 1)^2 + (-3 - 5)^2} = \sqrt{(-1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65} \]
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Distance CD: \[ \sqrt{(0 - 4)^2 + (-3 - 3)^2} = \sqrt{(-4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \]
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Identifying rectangle vertices: The opposite sides must be equal in length:
- Sides AB and CD have the same length \(\sqrt{52}\).
- Sides AD and BC have the same length \(\sqrt{13}\).
Thus, the rectangle is defined by points A, B, D, and C.
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Finding the lengths of sides:
- Length AB (or CD) = \(\sqrt{52} = 2\sqrt{13}\).
- Length AD (or BC) = \(\sqrt{13}\).
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Computing the area: The area \(Area\) of the rectangle is given by: \[ Area = length \times width = (2\sqrt{13}) \times (\sqrt{13}) = 2 \cdot 13 = 26 , \text{units}^2. \]
Thus the area of the rectangle is 26 units².
The answer is: d. 26 units²