To find the perimeter of the polygon with the given coordinates, we will first identify the points and then calculate the lengths of the sides formed by each pair of consecutive points.
The given coordinates are:
- \( A(0, 2) \)
- \( B(2, 0) \)
- \( C(1, -2) \)
- \( D(-1, -2) \)
- \( E(-2, 0) \)
Now, let's calculate the lengths of the sides using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Length of side AB: \[ d_{AB} = \sqrt{(2 - 0)^2 + (0 - 2)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
-
Length of side BC: \[ d_{BC} = \sqrt{(1 - 2)^2 + (-2 - 0)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
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Length of side CD: \[ d_{CD} = \sqrt{(-1 - 1)^2 + (-2 - (-2))^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2.0 \]
-
Length of side DE: \[ d_{DE} = \sqrt{(-2 - (-1))^2 + (0 - (-2))^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
-
Length of side EA: \[ d_{EA} = \sqrt{(0 - (-2))^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
Now, we can sum all the side lengths to find the perimeter \( P \):
\[ P = d_{AB} + d_{BC} + d_{CD} + d_{DE} + d_{EA} \]
Calculating the total: \[ P \approx 2.8 + 2.2 + 2.0 + 2.2 + 2.8 = 12.0 \]
Thus, the perimeter of the polygon is approximately 12.0 units.