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Triangles; Area; Inequalities Quant Review 2nd Ed #157

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by runningguy » Thu Sep 19, 2013 9:00 am
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 > sqrt2
B) sqrt3/2 < y < sqrt2
C) sqrt2/3 < y < sqrt3/2
D) sqrt3/4 < y < sqrt2/3
E) y < sqrt3/4
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by theCodeToGMAT » Thu Sep 19, 2013 9:13 am
Z = Hypotenuse
X & y = other two sides
So,

x * y * 1/2 = 1
xy=2

One possible solution is x = sqrt[1/2] and y = sqrt[4]
so
y = sqrt[ x^2 + y^2]
= sqrt[ 1/4 + 16]
~ 4._ _

So, Answer is [spoiler][A][/spoiler]

What is the OA????
Last edited by theCodeToGMAT on Thu Sep 19, 2013 9:58 am, edited 1 time in total.
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by GMATGuruNY » Thu Sep 19, 2013 9:23 am
runningguy wrote: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 > sqrt2
B) sqrt3/2 < y < sqrt2
C) sqrt2/3 < y < sqrt3/2
D) sqrt3/4 < y < sqrt2/3
E) y < sqrt3/4
Plug in an easy triangle whose area is 1.
Let x=1 and y=2, so that the area of the triangle = (1/2)(1)(2) = 1.
Since it's possible that y=2, eliminate any answer choice that does not include y=2 in its range.
Eliminate B, C, D, and E.

The correct answer is A.
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by Brent@GMATPrepNow » Thu Sep 19, 2013 9:45 am
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 another 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
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by runningguy » Thu Sep 19, 2013 12:23 pm
Brent@GMATPrepNow wrote:
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 another 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
Brent,

Why does X need to be less than Y?
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by [email protected] » Thu Sep 19, 2013 1:02 pm
Hi runningguy,

X has to be less than Y because the prompt says so (in the first sentence).

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