Explanation
The only way for a right triangle to have two sides of
equal length is to be a 45-45-90 triangle. That means that the lengths of the
sides of this triangle are in the ratio
. On to the statements, separately first.
Statement (1) gives us the area of the triangle. Since the
two sides of length x are
perpendicular, they are valid base-height pair. Therefore,
. That allows us to solve for x (it's
). From there, we can determine the lengths of all
of the sides and hence the perimeter. Statement (1) is sufficient.
Statement (2) tells us that
, so we can solve for x and determine the perimeter of the
triangle. Statement (2) is therefore sufficient and the correct answer is (D).
We see in this question that special triangles, like
circles, are prone to sufficiency, because they are defined by a small number of
parameters.
Again, the correct answer is (D).
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