To find the perimeter of a rectangle, we first need to determine the lengths of its sides. The perimeter (P) is calculated using the formula:
\[ P = 2(\text{length} + \text{width}) \]
We will identify pairs of points that represent the vertices of the rectangle.
The given coordinates of the vertices are \( A(0, 0) \), \( B(4, 4) \), \( C(5, 3) \), and \( D(1, -1) \).
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Finding the lengths of the sides:
- Calculate the distances between points \( A(0, 0) \) and \( B(4, 4) \) (which will give us one side): \[ d(A, B) = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- Calculate the distances between points \( B(4, 4) \) and \( C(5, 3) \): \[ d(B, C) = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{1 + 1} = \sqrt{2} \]
- Calculate the distances between points \( C(5, 3) \) and \( D(1, -1) \): \[ d(C, D) = \sqrt{(1 - 5)^2 + (-1 - 3)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
- Calculate the distances between points \( D(1, -1) \) and \( A(0, 0) \): \[ d(D, A) = \sqrt{(0 - 1)^2 + (0 - (-1))^2} = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Identify pairs of equal sides:
- We have two sides of length \( 4\sqrt{2} \) and two sides of length \( \sqrt{2} \).
Now we compute the perimeter:
\[ P = 2(\text{length} + \text{width}) = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) \]
Calculating \( 5\sqrt{2} \):
\[ \sqrt{2} \approx 1.414, \quad 5\sqrt{2} \approx 5 \times 1.414 = 7.07 \]
Thus,
\[ P = 2 \times 7.07 \approx 14.14 \]
Finally, rounding to the nearest tenth gives:
\[ \boxed{14.1} \]
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