A right triangle has two legs of equal length. What is the perimeter,

Official 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 \(x:x:\sqrt{2}\). 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, bh/2 = x^2/2 = 4. That allows us to solve for x (it's \(2\sqrt{2}\)). 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 \(x\sqrt{2}=4\), 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).