To find the area of a rectangle given its vertices, we can first verify that the given points indeed form a rectangle and then use the coordinates to compute the area.
The vertices given are:
- A(-3, -1)
- B(1, 5)
- C(4, 3)
- D(0, -3)
Step 1: Verify the structure of the rectangle
We need to check if the lengths of the sides formed by the points are perpendicular and equal in pairs.
- Find the distances between the points:
- Distance AB: \[ AB = \sqrt{(1 - (-3))^2 + (5 - (-1))^2} = \sqrt{(1 + 3)^2 + (5 + 1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]
- Distance BC: \[ BC = \sqrt{(4 - 1)^2 + (3 - 5)^2} = \sqrt{(3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
- Distance CD: \[ CD = \sqrt{(4 - 0)^2 + (3 - (-3))^2} = \sqrt{(4)^2 + (3 + 3)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]
- Distance DA: \[ DA = \sqrt{((-3) - 0)^2 + ((-1) - (-3))^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]
Step 2: Determine if opposite sides are equal and perpendicular
From the above calculations, we can conclude:
- AB and CD are both \(2\sqrt{13}\)
- BC and DA are both \(\sqrt{13}\)
Step 3: Calculate the area of the rectangle
To compute the area of the rectangle, we can use the length of two adjacent sides. Based on the distances:
- One pair of opposite sides is \(2\sqrt{13}\)
- The other pair of opposite sides is \(\sqrt{13}\)
Now we calculate the area: \[ \text{Area} = \text{Length} \times \text{Width} = (2\sqrt{13}) \times (\sqrt{13}) = 2 \times 13 = 26 \]
Conclusion
Thus, the exact area of the rectangle formed by the points A, B, C, and D is \( \boxed{26} \).
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