Problem Solving

A certain right triangle has sides of length x, y, and z, where x < y < z.lf the area of this triangular region is 1, which of the following indicates all of the possible values of y?

(A) y > √2
(B) √3/2 < y < √2
(C) √2/3 < y < √3/2
(D) √3/4 < y < √2/3
(E) y < √3/4

Here's one approach:

Since the triangle is a right triangle, the longest side will be the hypotenuse.
This means the side with length z is the hypotenuse, and the sides with lengths x and y are the legs.

Since the legs must meet at a right angle (since this is a right triangle), we can let one side (x or y) be the base of the triangle and the other side (x or y) can be the height.

We know that area = (base)(height)/2, and we're told the area of this particular triangle is 1.
So, we can conclude that (x)(y)/2 = 1
We can rewrite this as xy = 2

Now we also need to keep in mind that x < y

Given this, we could have x=0.1 and y=20, or we could have x=0.001 and y=2000, and so on.

As you can see, we can make y as large as we wish.
So, we can eliminate answer choices B, C, D and E since they all limit the upper value of y.

Answer = A

Cheers,
Brent