Triangle A has one side of length x. If (x^8) ^ 1/2 = 81 , what is the perimeter of Triangle A?

1) Triangle A has sides whose lengths are consecutive integers
2) Triangle A is NOT a right triangle



OE - By simplifying the equation given in the question stem, we can solve for x as follows:

(x^8) ^ 1/2 = 81

x^4 = 81

x = 3


Thus, we know that one side of Triangle A has a length of 3.

Statement (1) tells us that Triangle A has sides whose lengths are consecutive integers. Given that one of the

sides of Triangle A has a length of 3, this gives us the following possibilities: (1, 2, 3) OR (2, 3, 4) OR (3, 4,

5).

However, the first possibility is NOT a real triangle, since it does not meet the following condition, which is

true for all triangles: The sum of the lengths of any two sides of a triangle must always be greater than the

length of the third side. Since 1 + 2 is not greater than 3, it is impossible for a triangle to have side lengths

of 1, 2 and 3.

Thus, Statement (1) leaves us with two possibilities. Either Triangle A has side lengths 2, 3, 4 and a perimeter of

9 OR Triangle A has side lengths 3, 4, 5 and a perimeter of 12. Since there are two possible answers, Statement (1)

is not sufficient to answer the question.

Statement (2) tells us that Triangle A is NOT a right triangle. On its own, this is clearly not sufficient to

answer the question, since there are many non-right triangles that can be constructed with a side of length 3.

Taking both statements together, we can determine the perimeter of Triangle A.

From Statement (1) we know that Triangle A must have side lengths of 2, 3, and 4 OR side lengths of 3, 4, and 5.

Statement (2) tells us that Triangle A is not a right triangle; this eliminates the possibility that Triangle A has

side lengths of 3, 4, and 5 since any triangle with these side lengths is a right triangle (this is one of the

common Pythagorean triples). Thus, the only remaining possibility is that Triangle A has side lengths of 2, 3, and

4, which yields a perimeter of 9.

The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.