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# Is the length of a side of equilateral triangle E less than

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Re: Is the length of a side of equilateral triangle E less than [#permalink]
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bparrish89 wrote:
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{x\frac{\sqrt{3}}{2}}{y\sqrt{2}}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

Bunuel - Could you explain how you simplified the ratio in statement #2 to get to x/y = 4/3?

The second statement states the ratio as 2√3 : 3√2 &, the calculated ratio is x√3/2 : y√2. Now if these two ratios are same, we just need to simplify the equation, which gives the ratio of x:y to 4:3.
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
bparrish89 wrote:
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{x\frac{\sqrt{3}}{2}}{y\sqrt{2}}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

Bunuel - Could you explain how you simplified the ratio in statement #2 to get to x/y = 4/3?

Sure.

$$\frac{(\frac{x\sqrt{3}}{2})}{(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$;

$$\frac{x\sqrt{3}}{2(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$;

Divide both sides by $$\frac{\sqrt{3}}{\sqrt{2}}$$: $$\frac{x}{2y}=\frac{2}{3}$$;

Multiply by 2: $$\frac{x}{y}=\frac{4}{3}$$.

Hope it's clear.
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
bparrish89 wrote:
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{x\frac{\sqrt{3}}{2}}{y\sqrt{2}}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

Bunuel - Could you explain how you simplified the ratio in statement #2 to get to x/y = 4/3?

If equilateral triangle has height 2square root 3.. that means its all sides will be 4..
and if diagonal of square is 3 square root2 that means square has all sides 3.

we got No ! equilateral triangle length is greater than square's length
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
Bunuel wrote:
bparrish89 wrote:
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{x\frac{\sqrt{3}}{2}}{y\sqrt{2}}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

Bunuel - Could you explain how you simplified the ratio in statement #2 to get to x/y = 4/3?

Sure.

$$\frac{(\frac{x\sqrt{3}}{2})}{(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$;

$$\frac{x\sqrt{3}}{2(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$;

Divide both sides by $$\frac{\sqrt{3}}{\sqrt{2}}$$: $$\frac{x}{2y}=\frac{2}{3}$$;

Multiply by 2: $$\frac{x}{y}=\frac{4}{3}$$.

Hope it's clear.

That's exactly what I was looking for. Thanks!
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{(x\frac{\sqrt{3}}{2})}{(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

In 2 above, can you tell me how you got y√2?
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
X017in wrote:
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{(x\frac{\sqrt{3}}{2})}{(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

In 2 above, can you tell me how you got y√2?

y is the length of a side of square F. Now, the diagonal of a square is the hypotenuse of a right isosceles triangle made by the sides:
Attachment:

square.jpg [ 10.18 KiB | Viewed 20031 times ]

Therefore by Pythagorean theorem $$y^2+y^2=diagonal^2$$ --> $$2y^2=diagonal^2$$ --> $$diagonal=y\sqrt{2}$$.

Hope it's clear.
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
Could you please explain how did you get the calculated height of the equilateral triangle in statement 2?
Thanks
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
millopezle wrote:
Could you please explain how did you get the calculated height of the equilateral triangle in statement 2?
Thanks

hi millopezle
Is the length of a side of equilateral triangle E less than the length of a side of square F?

(1) The perimeter of E and the perimeter of F are equal.
(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2

it is giving us the ratio of height of triangle E to the diagonal of square F as 2√3 : 3√2...
since its a ratio ,we can multiply both by x, although we dont require that because final answer is also a ratio...
from height of triangle , we can get its side by formula.. h=side1*√3/2...
from diagonal of square we can get side by formula... diagonal=√2*side2
what you require is side1/side2= 2h/√3*dia/√2=2/√6*h/dia=2/√6*2√3/3√2=2/3...
so we have the ratio as 2/3..
so we can say side of square >side of tri... sufficient
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{(x\frac{\sqrt{3}}{2})}{(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

I have a Question, why do we assume that sides of these figures (triangle and square) are integers?
If its a non-integer then the entire answer changes to E.
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
Hi earnit,

You bring up a fair point - we don't have to assume that the side lengths are integers, but it makes dealing with the 'logic' behind this question easier. As it stands, using non-integers will NOT change the answer to the question, but certain DS questions will require that you consider non-integer values, so it's a good idea to keep them in mind.

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Re: Is the length of a side of equilateral triangle E less than [#permalink]
EMPOWERgmatRichC wrote:
Hi earnit,

You bring up a fair point - we don't have to assume that the side lengths are integers, but it makes dealing with the 'logic' behind this question easier. As it stands, using non-integers will NOT change the answer to the question, but certain DS questions will require that you consider non-integer values, so it's a good idea to keep them in mind.

GMAT assassins aren't born, they're made,
Rich

Thank you.
I accidentally also missed the fact that changing the values to non-integer will not affect the ratio and the answer.
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
Bunuel wrote:
Is the length of a side of equilateral triangle E less than the length of a side of square F?

Let x be the length of a side of equilateral triangle E and y be the length of a side of square F.
Question: is x>y?

(1) The perimeter of E and the perimeter of F are equal --> 3x=4y --> x/y=4/3 --> x>y. Sufficient.

(2) The ratio of the height of triangle E to the diagonal of square F is 2√3 : 3√2 --> the height of triangle E is $$x\frac{\sqrt{3}}{2}$$ and the diagonal of square F is $$y\sqrt{2}$$ --> ratio: $$\frac{(x\frac{\sqrt{3}}{2})}{(y\sqrt{2})}=\frac{2\sqrt{3}}{3\sqrt{2}}$$ --> x/y=4/3 --> x>y. Sufficient.

I find myself a bit off-track here :/

The question says "is the length of a side of the equilateral triangle less than the length of a side of the square?"
If we assume the length of a side of the equilateral triangle = x
and the length of a side of the square = y; aren't we looking for if x<y as explicitly stated in the question? Why did we look for if x>y?

Regards,
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
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Re: Is the length of a side of equilateral triangle E less than [#permalink]
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