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

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

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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 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

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

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 \(\frac{xy}{2}=\frac{x*1,000,000}{2}=1\) --> \(x=\frac{2}{1,000,000}\).

Hope it helps.
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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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New post 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 \(\frac{2}{y}<y\) --> \(2<y^2\) --> \(\sqrt{2}<y\).

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 \(\frac{xy}{2}=\frac{x*1,000,000}{2}=1\) --> \(x=\frac{2}{1,000,000}\).

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 ?

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

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New post 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.

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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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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)

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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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New post 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?

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

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

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 \(\frac{xy}{2}=\frac{x*1,000,000}{2}=1\) --> \(x=\frac{2}{1,000,000}\).

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.

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

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 \(\frac{xy}{2}=\frac{x*1,000,000}{2}=1\) --> \(x=\frac{2}{1,000,000}\).

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.

Please confirm


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

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

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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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New post 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.

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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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New post 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}

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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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New post 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 ....
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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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New post 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 \(>\sqrt{2}\).

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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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New post 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

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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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New post 11 Sep 2017, 13:54
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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:
Image

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:
[Reveal] Spoiler:
A


Cheers,
Brent
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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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New post 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.

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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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New post 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. \(y < \frac {\sqrt {3}}{4}\)


The answer is A

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