To find the number of units of tile Alex will need to surround his pool, we need to calculate the perimeter of the quadrilateral ABCD.
First, we find the lengths of all sides:
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Length of AB: Given as 2.24 units.
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Length of BC: To find this side, we'll calculate the distance between points B (2, 4) and C (4, 0).
Using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] we have: \[ d = \sqrt{(4 - 2)^2 + (0 - 4)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \text{ units} \]
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Length of CD: To find this length, we calculate the distance between C (4, 0) and D (2, -1). \[ d = \sqrt{(2 - 4)^2 + (-1 - 0)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.24 \text{ units} \]
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Length of DA: To find this length, we calculate the distance between D (2, -1) and A (0, 3). \[ d = \sqrt{(0 - 2)^2 + (3 - (-1))^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \text{ units} \]
Now we can calculate the total perimeter \(P\) of quadrilateral ABCD:
\[ P = AB + BC + CD + DA \] \[ P = 2.24 + 4.47 + 2.24 + 4.47 \] \[ P = 2.24 + 2.24 + 4.47 + 4.47 = 4.48 + 8.94 = 13.42 \text{ units} \]
The total length of tile needed to surround his pool is approximately 13.42 units.
Rounding to the nearest hundredth, the answer is: \[ \boxed{13.42} \]
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