To find the perimeter of the polygon with the given coordinates, we first identify the coordinates for each point:
- A = (-1, -2)
- B = (-2, 0)
- C = (0, 2)
- D = (2, 0)
- E = (1, -2)
Next, we will calculate the distance between each pair of consecutive points to find the lengths of the sides of the polygon. We'll also include the distance between the last point E and the first point A to close the polygon.
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Now, let's calculate the distances:
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Distance between A and B: \[ d_{AB} = \sqrt{((-2) - (-1))^2 + (0 - (-2))^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
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Distance between B and C: \[ d_{BC} = \sqrt{(0 - (-2))^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
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Distance between C and D: \[ d_{CD} = \sqrt{(2 - 0)^2 + (0 - 2)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
-
Distance between D and E: \[ d_{DE} = \sqrt{(1 - 2)^2 + (-2 - 0)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
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Distance between E and A: \[ d_{EA} = \sqrt{((-1) - 1)^2 + (-2 - (-2))^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2.0 \]
Now we sum all the calculated distances to find the perimeter: \[ \text{Perimeter} = d_{AB} + d_{BC} + d_{CD} + d_{DE} + d_{EA} \] \[ \text{Perimeter} \approx 2.2 + 2.8 + 2.8 + 2.2 + 2.0 \approx 12.0 \]
Thus, the perimeter of the polygon is approximately 12.0.