I believe so, though I am not sure I can prove it.
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5, 12, 13 and 6, 8, 10
True.
No the area is not always larger than the perimeter. Ex. The area of a reectangle could be 4 feet. The width could be 4 while the length is 1. The perimeter total would be 10.
No. A rectangle of 1 x 3 has the same perimeter as a rectangle of 2 x 2, but the areas are different.