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Which of the following figures has the largest perimeter $(1 \text{ foot} = 12 \text{ inches})$

1. a square with a diagonal of $5$ feet
2. a rectangle with sides of $3$ feet and $4$ feet
3. an equilateral triangle with a side equal to $48$ inches
4. a regular hexagon whose longest diagonal is $6$ feet

(A) Diagonal $= \sqrt{2}\times side=5 feet \\ \implies side = \frac{5}{\sqrt{2}} feet$

Parameter $=4\times side = 4 \times \frac{5}{\sqrt{2}} = 10\sqrt{2} feet = 14.142 feet$

(B) $a=3 feet, b=4 feet$

Parameter $=2\times(a+b) = 2\times(3+4) = 2\times 7 = 14 feet$

(C) $side =48 inches = \frac{48}{12} = 4 feet$

Parameter $=3\times side = 3\times 4 = 12 feet$

(D) longest diagonal $=2\times side = 6 feet \\ \implies side = 3 feet$

Parameter $= 6\times side = 6 \times 3 = 18 feet$

Option D is correct.

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