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

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Math Expert
Joined: 02 Sep 2009
Posts: 59724
A certain right triangle has sides of length x, y and z, where x < y <  [#permalink]

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06 Jul 2017, 02:11
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55% (hard)

Question Stats:

59% (02:09) correct 41% (02:22) wrong based on 101 sessions

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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 z?

(A) z > 2
(B) √2 < z < 2
(C) √2 < z < √3
(D) 1 < z < √2
(E) z < 1

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

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06 Jul 2017, 02:12
1
Bunuel 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 z?

(A) z > 2
(B) √2 < z < 2
(C) √2 < z < √3
(D) 1 < z < √2
(E) z < 1

This is a modified question of the the following OG question: https://gmatclub.com/forum/a-certain-ri ... 68727.html
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Re: A certain right triangle has sides of length x, y and z, where x < y <  [#permalink]

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06 Jul 2017, 02:37
Bunuel 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 z?

(A) z > 2
(B) √2 < z < 2
(C) √2 < z < √3
(D) 1 < z < √2
(E) z < 1

Right triangle has sides x, y, z and z being the highest must be Hypotenuse

i.e. (1/2)*x*y = 1
i.e. x*y = 1
@$$y_{min}$$ , x must be maximum and z must be Minimum
and then first unacceptable solution will be
@x = y, i.e. $$y^2 = 2$$
@x = y, i.e. $$y = \sqrt{2}$$

@x=y. $$z_{min} = 2$$

but since x<y, therefore z must be greater than 2

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

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06 Jul 2017, 02:38
Bunuel 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 z?

(A) z > 2
(B) √2 < z < 2
(C) √2 < z < √3
(D) 1 < z < √2
(E) z < 1

ALTERNATIVELY

Take any random case

i.e. x = 1/8, y = 16 and calculate z i.e. z > 16

All options eliminated except the first option

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

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08 Jul 2017, 10:52
1

since we know that area is 1 and it is given that x < y < z for the right angled triangle, we can say that z is the longest triangle. x*y = 2. let's say y(2) > x(1), then z>2.
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Re: A certain right triangle has sides of length x, y and z, where x < y <  [#permalink]

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30 Jul 2017, 06:27
brs1cob wrote:

since we know that area is 1 and it is given that x < y < z for the right angled triangle, we can say that z is the longest triangle. x*y = 2. let's say y(2) > x(1), then z>2.

Hi brs1cob,

For any given triangle if two sides are given then the third side has to be less than the sum and greater than the difference of the two sides.
So in the current case since xy=2, if we take x =1, y=2, then the value of z can only be greater than 1 and less than 3.

Where am I going wrong?

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

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09 Aug 2019, 08:56
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Re: A certain right triangle has sides of length x, y and z, where x < y <   [#permalink] 09 Aug 2019, 08:56
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