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The length of each side of equilateral triangle M is 3 times the lengt

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The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 20 May 2019, 00:05
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FRESH GMAT CLUB TESTS QUESTION



The length of each side of equilateral triangle M is 3 times the length of each side of equilateral triangle N. What is the perimeter of equilateral triangle N ?

(1) The ratio of the area of equilateral triangle M to the area of equilateral triangle N is 9:1
(2) The sum of the circumferences of circles circumscribing triangles M and N is \(12\pi\)

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Re: The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 20 May 2019, 04:14
1) Ratio of area of triangle M to area of triangle N = 9:1= square of respective sides,
So, one can find the side of triangle M & triangle N & that is 3:1 respectively. Hence, we can find unique value (i.e. 3) of perimeter of both triangles. Sufficient.
2) With information in this statement, one cannot find the individual side of triangle M & or triangle N. Insufficient. Ans. A.

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Re: The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 20 May 2019, 04:43
Raxit85 wrote:
1) Ratio of area of triangle M to area of triangle N = 9:1= square of respective sides,
So, one can find the side of triangle M & triangle N & that is 3:1 respectively. Hence, we can find unique value (i.e. 3) of perimeter of both triangles. Sufficient.

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Don't we have this information in the description of the problem?
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The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 20 May 2019, 04:59
According to me, length of each side of triangle M = 3 X length of each side of triangle N, means side of M:N could be 3:1/6:2/9:3. Hence, we can't surely find unique value of perimeter of either triangle. But with above mentioned formula, we can get the exact side of triangles. Therefore, we can get exact value of perimeter of the triangles.

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The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 20 May 2019, 06:02
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



The length of each side of equilateral triangle M is 3 times the length of each side of equilateral triangle N. What is the perimeter of equilateral triangle N ?

(1) The ratio of the area of equilateral triangle M to the area of equilateral triangle N is 9:1
(2) The sum of the circumferences of circles circumscribing triangles M and N is \(12\pi\)


(I) just repeats the information already provided that is the sides are in ratio 1:3..

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If sides of two similar triangles are in some ratio, the ratio of areas will be square of that ratio. Here 1:3 will mean areas will have 1^2:3^2 or 1:9.

(II) gives us info to work on. We can find the radius in each case in terms of sides a and 3a and adding these will give us 12pi. Since the variable is only one that is a, you will get the value.
Side a will give altitude √3*a/2, and 2/3 of this altitude will give the incenter or the radius, so radius is (√3)a/2*(2/3)=√3a/3, so other radius is √3(3a)/3=√3*a
Sum of circumference is \(2\pi*(\frac{√3a}{3}+√3a)=12\pi.....\)
\(\pi\) will get cancelled out and we can get a..
Suff

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Re: The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 20 May 2019, 06:13
Hi, Chetan2u, can you please elaborate statement 2 with diagram?

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Re: The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 20 May 2019, 07:19
Raxit85 wrote:
Hi, Chetan2u, can you please elaborate statement 2 with diagram?

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The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 22 May 2019, 02:38
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



The length of each side of equilateral triangle M is 3 times the length of each side of equilateral triangle N. What is the perimeter of equilateral triangle N ?

(1) The ratio of the area of equilateral triangle M to the area of equilateral triangle N is 9:1
(2) The sum of the circumferences of circles circumscribing triangles M and N is \(12\pi\)



1) Ratio of length of sides= 3:1, therefore ratio of areas is 3^2:1^2 or 9:1. Repetition of info given in question. Insufficient.
2) Circumference of a circle is \(2\pi r\), As the highest power of r is one and \(2\pi\) is constant which will cancel out in a fraction, the ratio of circumference of triangles is equal to ratio of the circumscribed radii of the equilateral triangles, the radii are proportional to the sides of equilateral triangles (explanation given below). Hence, circumferences of circles circumscribing triangles N= \(12\pi\) * \(\frac{1}{4}\)= \(3\pi\). From this we can easily calculate the side of equilateral triangle and 3 times side is perimeter of equilateral triangle N. Sufficient.

Answer is B.

For problem solving questions, the relationship between side of an equilateral triangle and radius of circumscribed circle is given below:
For an equilateral triangle with side S, the median is \(\frac{\sqrt{3}S}{2}\) and \(\frac{1}{3}\) of median is radius of circumscribed circle. Therefore, radius of circumscribed circle is \(\frac{S}{\sqrt{3}}\)

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Re: The length of each side of equilateral triangle M is 3 times the lengt  [#permalink]

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New post 19 Oct 2019, 07:39
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



The length of each side of equilateral triangle M is 3 times the length of each side of equilateral triangle N. What is the perimeter of equilateral triangle N ?

(1) The ratio of the area of equilateral triangle M to the area of equilateral triangle N is 9:1
(2) The sum of the circumferences of circles circumscribing triangles M and N is \(12\pi\)


\(M_{side}=3y…N_{side}=y…N_{perimeter}=3(y)=3y=M_{side}\)

(1) The ratio of the area of equilateral triangle M to the area of equilateral triangle N is 9:1 insufic.

\(\frac{M_{area}}{N_{area}}=9…\frac{(3y)^2√3/4}{(y)^2√3/4}=9…\frac{9y^2}{y^2}=9…9=9…y=?\)

(2) The sum of the circumferences of circles circumscribing triangles M and N is \(12\pi\) sufic.

\(Circle_M…30:60:90=altitude,side/2,radius=a:a√3:2a…\)
\(M_{side}/2=(3y)/2=a√3…a=3y/(2√3)…M_{radius}=2[3y/(2√3)]=3y/√3\)

\(Circle_N…30:60:90=a:a√3:2a…\)
\(N_{side}/2=(y)/2=a√3…a=y/(2√3)…N_{radius}=2[y/(2√3)]=y/√3\)

\(2π(M_{radius}+N_{radius})=12π…(3y/√3+y/√3)=6…4y√3/3=6…4y√3=18…y√3=9/2…y=4.5/√3\)

Answer (B)
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Re: The length of each side of equilateral triangle M is 3 times the lengt   [#permalink] 19 Oct 2019, 07:39
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