If a rectangular region has perimeter P inches and area A square inches, is the region square?
(1) P = 4/3*A
(2) P = 4√A
Statement 1: P = (4/3)A
Case 1: A=9
If the rectangle is a square with an area of 9, then s=3 and P = 4s = 12.
This works, since statement 1 requires that P = (4/3)(A) = (4/3)(9) = 12.
Thus:
In Case 1, the rectangle is a square with a side of 3.
Case 2: A=81
If the rectangle is a square with an area of 81, then s=9 and P = 4s = 36.
This doesn't work, since statement 1 requires that P = (4/3)(A) = (4/3)(81) = 108.
Thus:
In Case 2, the rectangle is NOT a square.
INSUFFICIENT.
Statement 2: P = 4√A
Case 1: A=9
If the rectangle is a square with an area of 9, then s=3 and P = 4s = 12.
This works, since statement 2 requires that P = 4√A = 4√9 = 12.
Thus:
In Case 1, the rectangle is a square with a side of 3.
Case 2: A=81
If the rectangle is a square with an area of 81, then s=9 and P = 4s = 36.
This works, since statement 2 requires that P = 4√A = 4√81 = 36.
Thus:
In Case 2, the rectangle is a square with a side of 9.
The cases above illustrate that, in statement 2, the rectangle must be a square.
SUFFICIENT.
The correct answer is
B.
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