To calculate the perimeter of the rectangle given the vertices at the coordinates (0,0), (4,4), (5,3), and (1,−1), we first need to verify if these points form a rectangular shape.
Step 1: Verify that the figure is a rectangle.
A rectangle has opposite sides equal and all angles equal to 90 degrees. We can check this by calculating the lengths of the sides and ensuring that the diagonals are equal.
Step 2: Calculate the lengths of the sides formed by the vertices.
Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \):
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Distance between (0,0) and (4,4): \[ d_1 = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance between (4,4) and (5,3): \[ d_2 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Distance between (5,3) and (1,-1): \[ d_3 = \sqrt{(1 - 5)^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Distance between (1,-1) and (0,0): \[ d_4 = \sqrt{(0 - 1)^2 + (0 - (-1))^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
From these calculations, we see that the lengths of opposite sides are equal:
- Two sides of length \( 4\sqrt{2} \)
- Two sides of length \( \sqrt{2} \)
Step 3: Calculate the perimeter of the rectangle.
The perimeter \( P \) of a rectangle is given by: \[ P = 2(\text{length} + \text{width}) \]
Substituting the side lengths:
- Length \( = 4\sqrt{2} \)
- Width \( = \sqrt{2} \)
Calculating the perimeter: \[ P = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) = 10\sqrt{2} \]
Step 4: Approximate the value.
Using \( \sqrt{2} \approx 1.414 \): \[ P \approx 10 \cdot 1.414 = 14.14 \]
Final Answer: Rounding to the nearest tenth, the perimeter of the rectangle is approximately \( \boxed{14.1} \).
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