To calculate the perimeter of the polygon formed by the coordinates \((0,8)\), \((6,5)\), \((6,-1)\), \((-2,-4)\), and \((-5,3)\), we need to find the lengths of each side of the polygon and then sum those lengths.
We use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) for each pair of consecutive points.
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From \((0,8)\) to \((6,5)\): \[ d_1 = \sqrt{(6 - 0)^2 + (5 - 8)^2} = \sqrt{6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.71 \]
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From \((6,5)\) to \((6,-1)\): \[ d_2 = \sqrt{(6 - 6)^2 + (-1 - 5)^2} = \sqrt{0 + (-6)^2} = \sqrt{36} = 6 \]
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From \((6,-1)\) to \((-2,-4)\): \[ d_3 = \sqrt{(-2 - 6)^2 + (-4 + 1)^2} = \sqrt{(-8)^2 + (-3)^2} = \sqrt{64 + 9} = \sqrt{73} \approx 8.54 \]
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From \((-2,-4)\) to \((-5,3)\): \[ d_4 = \sqrt{(-5 + 2)^2 + (3 + 4)^2} = \sqrt{(-3)^2 + (7)^2} = \sqrt{9 + 49} = \sqrt{58} \approx 7.62 \]
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From \((-5,3)\) to \((0,8)\): \[ d_5 = \sqrt{(0 + 5)^2 + (8 - 3)^2} = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} \approx 7.07 \]
Now, we add all the lengths together to find the perimeter: \[ \text{Perimeter} \approx d_1 + d_2 + d_3 + d_4 + d_5 \] \[ \text{Perimeter} \approx 6.71 + 6 + 8.54 + 7.62 + 7.07 \approx 35.94 \]
Rounding to the nearest whole number, the perimeter of the polygon is approximately 36 units.