To find the perimeter of the rectangle given its vertices at coordinates (0,0), (4,4), (5,3), and (1,-1), we first need to confirm whether these points actually form a rectangle.
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Determining the Distances (Sides of the Rectangle)
We can calculate the lengths of the sides by finding the distance between each pair of adjacent vertices. Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]-
Between (0, 0) and (4, 4): \[ d_1 = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
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Between (4, 4) and (5, 3): \[ d_2 = \sqrt{(5 - 4)^2 + (3 - 4)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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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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Between (1, -1) and (0, 0): \[ d_4 = \sqrt{(0 - 1)^2 + (0 - (-1))^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
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Confirming Rectangle Properties
In a rectangle, opposite sides are equal, and adjacent sides are perpendicular. Here, we have:- Two sides of length \(4\sqrt{2}\)
- Two sides of length \(\sqrt{2}\)
Since the sides are not only equal in pairs but also appear to meet at right angles (inferred from the different distances), we can deduce that these points do form a rectangle.
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Calculating the Perimeter
The perimeter \(P\) of a rectangle is given by: \[ P = 2(\text{Length} + \text{Width}) = 2(4\sqrt{2} + \sqrt{2}) = 2(5\sqrt{2}) = 10\sqrt{2} \] -
Numerical Calculation
Next, let's evaluate \(10\sqrt{2}\) numerically: \[ \sqrt{2} \approx 1.414 \] \[ 10\sqrt{2} \approx 10 \times 1.414 = 14.14 \] -
Final Rounding
Rounding to the nearest tenth gives us: \[ \text{Perimeter} \approx 14.1 \]
Thus, the perimeter of the rectangle is approximately 14.1 units.