To find the perimeter of the polygon formed by the given coordinates, we first need to calculate the distances between each pair of consecutive points, and then sum those distances.
The coordinates are:
- A (-1, -2)
- B (-2, 0)
- C (0, 2)
- D (2, 0)
- E (1, -2)
We will use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate the distances between consecutive points
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Distance AB: \[ d_{AB} = \sqrt{((-2) - (-1))^2 + (0 - (-2))^2} = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
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Distance BC: \[ d_{BC} = \sqrt{(0 - (-2))^2 + (2 - 0)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
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Distance CD: \[ d_{CD} = \sqrt{(2 - 0)^2 + (0 - 2)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8 \]
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Distance DE: \[ d_{DE} = \sqrt{(1 - 2)^2 + (-2 - 0)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.2 \]
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Distance EA: \[ d_{EA} = \sqrt{((-1) - 1)^2 + ((-2) - (-2))^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2.0 \]
Step 2: Sum the distances to find the perimeter
Now, we will sum all the calculated distances:
\[ \text{Perimeter} = d_{AB} + d_{BC} + d_{CD} + d_{DE} + d_{EA} \]
Substituting the approximate values:
\[ \text{Perimeter} \approx 2.2 + 2.8 + 2.8 + 2.2 + 2.0 = 12.0 \]
Final Result
The perimeter of the polygon is approximately 12.0 units.
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