value of y

[gmatmachoman]

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


A. y > ROOT2
B. ROOT3/2 < y < ROOT2
C. ROOT2/3 < y < ROOT3/2
D. ROOT3/4 < y < ROOT2/3
E. y < ROOT3/4

[Stuart@KaplanGMAT]

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


A. y > ROOT2
B. ROOT3/2 < y < ROOT2
C. ROOT2/3 < y < ROOT3/2
D. ROOT3/4 < y < ROOT2/3
E. y < ROOT3/4

Since x and y are the legs (i.e. the non-hypotenuse sides) of the triangle, we can assign x and y as the base and height.

So:

Area = 1/2 * base * height

1 = 1/2 * x * y

2 = x * y

If x=y, then y would be root2. However, we know that y > x, so it must be true that y > root2.

Only A has y > root2 as a boundary... pick A!

(As an aside, there's no upward bound on y, since x could be a tiny little fraction and y could be huge; for example, we could pick:

x = 1/10000, giving us:

2 = (1/10000) * y

20000 = y.)

[uwhusky]

Is it A?

xy/2 = 1, xy = 2.

x = 2/y.

2/y < y < z

2 < y^2 < zy

√2 < y < √zy

I spent so much time trying to figure this question out using 30/60/90...Shoosh.

[Brent@GMATPrepNow]

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


A. y > ROOT2
B. ROOT3/2 < y < ROOT2
C. ROOT2/3 < y < ROOT3/2
D. ROOT3/4 < y < ROOT2/3
E. y < ROOT3/4

There are infinitely many right triangles that have an area of 1.
So, one approach is to find a triangle that meets the given conditions, and see what conclusions we can draw.

Here's one such right triangle:


This meets the conditions that the area is 1 AND x < y < z
With this triangle, y = 4

When we check the answer choices, only one (answer choice A) allows for y to equal 4

Answer: A

Cheers,
Brent

[Scott@TargetTestPrep]

A certain right triangle has sides of length x, y, and z, where x < y < z.
If 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

We see that x and y must be the legs of the triangle and z must be the hypotenuse. We also see that x and y are the base and altitude of the right triangle. Using the formula for the area of a triangle: A = ½ bh, we can create the equation:

xy/2 = 1

xy = 2

x = 2/y

We also know that x < y < z.

Substituting, we have:

2/y < y

2 < y^2

√2 < y

Answer: A