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A certain right triangle has sides of length x, y, and z, wh

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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) \(y<\frac{\sqrt{3}}{4}\)

Problem Solving
Question: 157
Category: Geometry; Algebra Triangles; Area; Inequalities
Page: 83
Difficulty: 600


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

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SOLUTION

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 \(\frac{2}{y}<y\) --> \(2<y^2\) --> \(\sqrt{2}<y\).

Answer: A.

Also note that max value of \(y\) is not limited at all. For example \(y\) can be \(1,000,000\) and in this case \(\frac{xy}{2}=\frac{x*1,000,000}{2}=1\) --> \(x=\frac{2}{1,000,000}\).
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A certain right triangle has sides of length x, y, and z, wh [#permalink]

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New post 20 Nov 2015, 13:11
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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) \(y<\frac{\sqrt{3}}{4}\)


This was a tough problem for me, to be honest I couldn't find a solution like Bunuel did, but I have tested aswer choices to derive at the correct answer.

We know that \(x*y=2\) actually let's square this expression \(x^2*y^2=4\) so let's test the values given in the answer choices with Min/Max approach (we ca also square values in the answer choices, as sides of a triangle can not be -ve)

(B) so \(y^2\) can be max 2 and \(x^2\)= max \(\frac{3}{4}\)--> multiply \(x^2*y^2\)=1,5. So B is out because we need a 4 when \(x^2 and y^2\) are multiplied
(C) \(y^2\) max = \(\frac{3}{4}\), \(x^2\) max=2/9 --> \(\frac{2}{9}*\frac{3}{4} < 4\) , C is out
(D) \(\frac{2}{9}*\frac{3}{16} < 4\)
(E) \(y^2\) < \(\frac{3}{16}\), so we know that y > x, thus when y < \(\frac{3}{16}\) \(x^2\) is also < \(\frac{3}{16}\) and their product is < 4, E is also out and we are left with Answer Choice A
(A)
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Re: A certain right triangle has sides of length x, y, and z, wh [#permalink]

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New post 27 Apr 2016, 16:59
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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) \(y<\frac{\sqrt{3}}{4}\)

Thank you![/textarea]


Hi,
from the area of triangle we know x*y = 2.
So there is one possibility that x could be 1 and y could be 2, but none of the answer choices ,except A confirms to y =2 possibility.

So A is our answer.
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A certain right triangle has sides of length x, y, and z, wh [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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) \(y<\frac{\sqrt{3}}{4}\)



You can think about it in terms of transition points too.

x < y< z so this means that x and y are the legs of the right triangle and z is the hypotenuse.
The area of the triangle will be (1/2)*xy = 1
xy = 2

Now, if x = y, then both x and y would be equal to \(\sqrt{2}\).
But y is greater than x, so y would be at least a slight bit greater than \(\sqrt{2}\) and x would be a slight bit less than \(\sqrt{2}\). In all options other than (A), y takes values less than 1.414.
Answer (A)
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A certain right triangle has sides of length x, y, and z, wh [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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) \(y<\frac{\sqrt{3}}{4}\)



First step will be to breakdown the options into recognizable decimal representations (assuming \(\sqrt{2} \approx 1.4\), \(\sqrt{3} \approx 1.7\))

A) y>1.4
B) 0.8<y<1.4
C) 0.5<y<0.8
D) 0.4<y<0.5
E) y<0.4

We are given that x<y<z and that 0.5*x*y=1 --> x*y=2

Now from the relation xy=2 --> go back to the options and test for y=1. You get x=2 but we are given that x<y ---> y MUST be > \(\approx\)1.4 such that x < y

For any value of y < 1.4 , you will end up getting x>y (try with y=0.5 or 0.75 etc).

Only A satisfies this condition and is hence the correct answer.

Hope this helps.
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Re: A certain right triangle has sides of length x, y, and z, wh [#permalink]

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Re: A certain right triangle has sides of length x, y, and z, wh   [#permalink] 02 May 2017, 21:42
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