To find the perimeter of the four-sided figure ABCD, we first need to calculate the lengths of the sides AB, BC, CD, and DA using the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculate AB:
- A(-4, 3) and B(0, 2) \[ AB = \sqrt{(0 - (-4))^2 + (2 - 3)^2} = \sqrt{(0 + 4)^2 + (2 - 3)^2} = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.12 \]
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Calculate BC:
- B(0, 2) and C(2, 4) \[ BC = \sqrt{(2 - 0)^2 + (4 - 2)^2} = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 \]
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Calculate CD:
- C(2, 4) and D(0, -3) \[ CD = \sqrt{(0 - 2)^2 + (-3 - 4)^2} = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53} \approx 7.28 \]
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Calculate DA:
- D(0, -3) and A(-4, 3) \[ DA = \sqrt{((-4) - 0)^2 + (3 - (-3))^2} = \sqrt{(-4)^2 + (3 + 3)^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21 \]
Now, we sum the lengths of all the sides to find the perimeter \(P\):
\[ P = AB + BC + CD + DA \] \[ P \approx 4.12 + 2.83 + 7.28 + 7.21 \] \[ P \approx 21.44 \]
Calculating this gives us:
\[ P \approx 21.4 \]
Thus, the perimeter of the shape is approximately 21.4.
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