It is currently 23 Oct 2017, 13:58

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

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

Author Message
TAGS:

### Hide Tags

Manager
Status: Current MBA Student
Joined: 19 Nov 2009
Posts: 128

Kudos [?]: 475 [4], given: 210

Concentration: Finance, General Management
GMAT 1: 720 Q49 V40
A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

16 Jan 2011, 16:01
4
KUDOS
58
This post was
BOOKMARKED
00:00

Difficulty:

75% (hard)

Question Stats:

60% (01:33) correct 40% (02:11) wrong based on 771 sessions

### HideShow timer Statistics

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. $$y > \sqrt {2}$$

B. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

C. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

D. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

E. $$y < \frac {\sqrt {3}}{4}$$
[Reveal] Spoiler: OA

Kudos [?]: 475 [4], given: 210

Math Expert
Joined: 02 Sep 2009
Posts: 41913

Kudos [?]: 129502 [30], given: 12201

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

16 Jan 2011, 16:12
30
KUDOS
Expert's post
29
This post was
BOOKMARKED
tonebeeze 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. $$y > \sqrt {2}$$

b. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

c. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

d. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

e. $$y < \frac {\sqrt {3}}{4}$$

The area of the triangle is $$\frac{xy}{2}=1$$ ($$x<y<z$$ means that hypotenuse is $$z$$) --> $$x=\frac{2}{y}$$. As $$x<y$$, then $$\frac{2}{y}<y$$ --> $$2<y^2$$ --> $$\sqrt{2}<y$$.

Also note that max value of $$y$$ is not limited at all. For example $$y$$ can be $$1,000,000$$ and in this case $$\frac{xy}{2}=\frac{x*1,000,000}{2}=1$$ --> $$x=\frac{2}{1,000,000}$$.

Hope it helps.
_________________

Kudos [?]: 129502 [30], given: 12201

Intern
Status: Focus is the Key
Joined: 07 Feb 2011
Posts: 9

Kudos [?]: 9 [0], given: 26

Concentration: Strategy, Technology
Schools: ISB
GMAT 1: 710 Q49 V38
GPA: 3.35
WE: Consulting (Telecommunications)
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

06 Aug 2011, 04:41
Bunuel wrote:
tonebeeze 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. $$y > \sqrt {2}$$

b. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

c. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

d. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

e. $$y < \frac {\sqrt {3}}{4}$$

The area of the triangle is $$\frac{xy}{2}=1$$ ($$x<y<z$$ means that hypotenuse is $$z$$) --> $$x=\frac{2}{y}$$. As $$x<y$$, then $$\frac{2}{y}<y$$ --> $$2<y^2$$ --> $$\sqrt{2}<y$$.

Also note that max value of $$y$$ is not limited at all. For example $$y$$ can be $$1,000,000$$ and in this case $$\frac{xy}{2}=\frac{x*1,000,000}{2}=1$$ --> $$x=\frac{2}{1,000,000}$$.

Hope it helps.

Dear Bunuel,

While solving the question ,I assumed it to be the special 90,60,30 triangle.
Am I wrong in following that approach ?

Kudos [?]: 9 [0], given: 26

Math Expert
Joined: 02 Sep 2009
Posts: 41913

Kudos [?]: 129502 [0], given: 12201

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

09 Feb 2012, 04:42
rohansharma wrote:
While solving the question ,I assumed it to be the special 90,60,30 triangle.
Am I wrong in following that approach ?

We are just told that the triangle is right, not that it's a special kind like 30-60-90 or 45-45-90.
_________________

Kudos [?]: 129502 [0], given: 12201

Manager
Status: MBA Aspirant
Joined: 12 Jun 2010
Posts: 172

Kudos [?]: 98 [0], given: 1

Location: India
WE: Information Technology (Investment Banking)
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

09 Feb 2012, 06:54
OA is A
since this is a rt angled triangle so z is th hypotnuse
and given xy = 2
so as x decreased y increases. Now if x is 1 then y is 2, when x is 1/2 y is 4.
Only option A supports this result.

Kudos [?]: 98 [0], given: 1

Manager
Joined: 27 Feb 2012
Posts: 94

Kudos [?]: 27 [1], given: 42

Concentration: General Management, Nonprofit
GMAT 1: 700 Q47 V39
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

13 Mar 2012, 15:11
1
KUDOS
Is there a reason we can completely ignore z in the inequality while solving for y? That is the only part I don't understand. (im rusty)

Kudos [?]: 27 [1], given: 42

Intern
Joined: 31 Dec 2012
Posts: 10

Kudos [?]: [0], given: 55

GMAT 1: 700 Q44 V42
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

28 Feb 2013, 09:55
Bunuel wrote:
$$x=\frac{2}{y}$$. As $$x<y$$, then $$\frac{2}{y}<y$$

Could you explain why that is?

Kudos [?]: [0], given: 55

Math Expert
Joined: 02 Sep 2009
Posts: 41913

Kudos [?]: 129502 [1], given: 12201

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

28 Feb 2013, 10:01
1
KUDOS
Expert's post
2flY wrote:
Bunuel wrote:
$$x=\frac{2}{y}$$. As $$x<y$$, then $$\frac{2}{y}<y$$

Could you explain why that is?

Just substitute x with $$\frac{2}{y}$$ in $$x<y$$ to get $$\frac{2}{y}<y$$.
_________________

Kudos [?]: 129502 [1], given: 12201

Director
Joined: 25 Apr 2012
Posts: 724

Kudos [?]: 850 [0], given: 724

Location: India
GPA: 3.21
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

01 Mar 2013, 02:21
rohansharma wrote:
Bunuel wrote:
tonebeeze 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. $$y > \sqrt {2}$$

b. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

c. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

d. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

e. $$y < \frac {\sqrt {3}}{4}$$

The area of the triangle is $$\frac{xy}{2}=1$$ ($$x<y<z$$ means that hypotenuse is $$z$$) --> $$x=\frac{2}{y}$$. As $$x<y$$, then $$\frac{2}{y}<y$$ --> $$2<y^2$$ --> $$\sqrt{2}<y$$.

Also note that max value of $$y$$ is not limited at all. For example $$y$$ can be $$1,000,000$$ and in this case $$\frac{xy}{2}=\frac{x*1,000,000}{2}=1$$ --> $$x=\frac{2}{1,000,000}$$.

Hope it helps.

Dear Bunuel,

While solving the question ,I assumed it to be the special 90,60,30 triangle.
Am I wrong in following that approach ?

Hi Bunuel,

The Q stem says that sides are x<y<z and it is a right angle triangle. So we can assume that it will be 30-60-90 triangle. Had it been 45-45-90 triangle then the 2 sides ie base and perpendicular would have been equal and therefore x=y and x,y<z

I guess it should be okay to assume that it is 30-60 -90 triangle.

_________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Kudos [?]: 850 [0], given: 724

Math Expert
Joined: 02 Sep 2009
Posts: 41913

Kudos [?]: 129502 [0], given: 12201

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

01 Mar 2013, 02:50
Expert's post
1
This post was
BOOKMARKED
mridulparashar1 wrote:
rohansharma wrote:
Bunuel wrote:
The area of the triangle is $$\frac{xy}{2}=1$$ ($$x<y<z$$ means that hypotenuse is $$z$$) --> $$x=\frac{2}{y}$$. As $$x<y$$, then $$\frac{2}{y}<y$$ --> $$2<y^2$$ --> $$\sqrt{2}<y$$.

Also note that max value of $$y$$ is not limited at all. For example $$y$$ can be $$1,000,000$$ and in this case $$\frac{xy}{2}=\frac{x*1,000,000}{2}=1$$ --> $$x=\frac{2}{1,000,000}$$.

Hope it helps.

Dear Bunuel,

While solving the question ,I assumed it to be the special 90,60,30 triangle.
Am I wrong in following that approach ?

Hi Bunuel,

The Q stem says that sides are x<y<z and it is a right angle triangle. So we can assume that it will be 30-60-90 triangle. Had it been 45-45-90 triangle then the 2 sides ie base and perpendicular would have been equal and therefore x=y and x,y<z

I guess it should be okay to assume that it is 30-60 -90 triangle.

Yes, if it were 45-45-90, then we would have that x=y<z. BUT, knowing that it's not a 45-45-90 right triangle does NOT mean that it's necessarily 30-60-90 triangle: there are numerous other right triangles. For example, 10-80-90, 11-79-90, 25-65-90, ...

Hope it's clear.
_________________

Kudos [?]: 129502 [0], given: 12201

Intern
Joined: 14 Sep 2013
Posts: 1

Kudos [?]: 1 [1], given: 0

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

05 Nov 2013, 11:04
1
KUDOS
tanviet wrote:
it is simple but hard enough to kill us

this is NOT og questions.

It is the Quant Review 2nd Ed. #157

Kudos [?]: 1 [1], given: 0

Manager
Joined: 07 May 2013
Posts: 108

Kudos [?]: 30 [0], given: 1

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

10 Nov 2013, 19:20
Buneul has quite literally owned this problem. Great solution!

Kudos [?]: 30 [0], given: 1

Intern
Joined: 12 Jun 2013
Posts: 3

Kudos [?]: [0], given: 9

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

15 Dec 2013, 08:04
i did some guesstimates to arrive at this choice

here we go -

if we assume this to be a isosceles right angled triangle then the area would be maximum.

and xy/2=1

y^2=1 (since it is an isosceles triangle)
y=Sq Root 2

now we know y>x and area =1; y has to be > sq root 2 and x < sq root 2

fortunately in this case, only one option had this range.

Kudos [?]: [0], given: 9

Senior Manager
Joined: 13 May 2013
Posts: 463

Kudos [?]: 198 [0], given: 134

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

16 Dec 2013, 14:17
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?

We are told that this is a right triangle which right off the bat tells me one of two things, either we need to solve with some variation of a^2 + b^2 = c^2 or that we can find the area with base*height.

Because this is a right triangle and x < y < z we know that z is the hypotenuse and that x is the shortest leg. The area = 1 so:

a=1/2 b*h
1=1/2 b*h
2=b*h.

Y is the second longest measurement in this right triangle which means it must be longer than x but shorter than z. If we run through a few possible combinations of a and b we see that there isn't a limit on the length of y so long as y*x = 2 and y<x. For example, x=1 and y = 4 and z can = 5. This means that there is no upward limit on the value of y so answer choice E is out. This also means that D, C and B are out as well because all contain upward limits on the value of y can be any number so long as y*x = 2 and y<x. Therefore, A is the only answer choice.

Answer: a. y > \sqrt {2}

Kudos [?]: 198 [0], given: 134

Director
Joined: 10 Mar 2013
Posts: 592

Kudos [?]: 463 [0], given: 200

Location: Germany
Concentration: Finance, Entrepreneurship
GMAT 1: 580 Q46 V24
GPA: 3.88
WE: Information Technology (Consulting)
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

23 Aug 2015, 07:02
I think I've made it more comlicated then it is...
xy=2, z^2=x^2+y^2
(x+y)^2=z^2+4 and just stucked at this point ....
_________________

When you’re up, your friends know who you are. When you’re down, you know who your friends are.

800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50
GMAT PREP 670
MGMAT CAT 630
KAPLAN CAT 660

Kudos [?]: 463 [0], given: 200

Current Student
Joined: 13 Feb 2011
Posts: 104

Kudos [?]: 48 [0], given: 3385

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

12 Feb 2016, 19:34
One other way that I noticed to solve this problem is to check the length of $$y$$ when $$x=y$$, i.e. 45,45,90. In that case $$x=y=\sqrt{2}$$, however as $$y>x$$, it'd always need to be $$>\sqrt{2}$$.

HTH

Kudos [?]: 48 [0], given: 3385

Manager
Joined: 23 Dec 2013
Posts: 235

Kudos [?]: 12 [0], given: 21

Location: United States (CA)
GMAT 1: 760 Q49 V44
GPA: 3.76
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

25 May 2017, 11:23
tonebeeze 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. $$y > \sqrt {2}$$

b. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

c. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

d. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

e. $$y < \frac {\sqrt {3}}{4}$$

Here we can solve this problem as follows

1/2 *xy = 1
xy =2

x = 2/y

2/y < y
2<y^2
sqrt(2) < y

Kudos [?]: 12 [0], given: 21

SVP
Joined: 12 Sep 2015
Posts: 1799

Kudos [?]: 2476 [0], given: 357

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

11 Sep 2017, 13:54
Expert's post
Top Contributor
tonebeeze 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. $$y > \sqrt {2}$$

b. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

c. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

d. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

e. $$y < \frac {\sqrt {3}}{4}$$

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

[Reveal] Spoiler:
A

Cheers,
Brent
_________________

Brent Hanneson – Founder of gmatprepnow.com

Kudos [?]: 2476 [0], given: 357

Intern
Joined: 29 Dec 2016
Posts: 3

Kudos [?]: [0], given: 7

Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

28 Sep 2017, 08:39
tonebeeze 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. $$y > \sqrt {2}$$

B. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

C. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

D. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

E. $$y < \frac {\sqrt {3}}{4}$$

Since y is the longest of all sides, we can conclude that it's hypotenuse. Hence, x & y are relevant for Area.
Since Area= 1, we can
1/2* x*y = 1.
Therefore, xy = 1.
Notice there could be any combination of x & y. What needs to be taken care of is x<y. Hence it could be 1 & 2, 1/2 & 4 etc.

Option A : It says Y> Sq.root 2
If we substitute same in the equation we get x also as sq.root 2. However, we know that x cannot equal to y. Hence, y can be anything greater than sq.root 2. This option satisfies the criteria. This doesn't leave any scope for Y value to differ as Y cannot be equal to less than sq.root of 2.

Kudos [?]: [0], given: 7

VP
Status: Learning
Joined: 20 Dec 2015
Posts: 1073

Kudos [?]: 69 [0], given: 535

Location: India
Concentration: Operations, Marketing
GMAT 1: 670 Q48 V36
GRE 1: 314 Q157 V157
GPA: 3.4
WE: Manufacturing and Production (Manufacturing)
Re: A certain right triangle has sides of length x, y, and z, where x < y [#permalink]

### Show Tags

29 Sep 2017, 11:22
tonebeeze 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. $$y > \sqrt {2}$$

B. $$\frac {\sqrt {3}} {2} < y < \sqrt {2}$$

C. $$\frac {\sqrt {2}} {3} < y < \frac {\sqrt {3}} {2}$$

D. $$\frac {\sqrt {3}} {4} < y < \frac {\sqrt {2}} {3}$$

E. $$y < \frac {\sqrt {3}}{4}$$

Area =1/2 x*y as it is right triangle and according to the condition x < y < z therefore the two legs of the triangle are x and y

so x*y=2

Now taking the condition x < y < z
Let us take x<y
Multiply both sides by y
we have xy<y^2
or 0<y^2-x*y
0<y^2 -2 or 0<(y-√2)*(y+√2)

As length can not be negative therefore we have y>√2
_________________

We are more often frightened than hurt; and we suffer more from imagination than from reality

Kudos [?]: 69 [0], given: 535

Re: A certain right triangle has sides of length x, y, and z, where x < y   [#permalink] 29 Sep 2017, 11:22
Display posts from previous: Sort by