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

Question

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

M36-62

Bunuel · Accepted Answer

Official Solution:

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 ?  
 
All equilateral triangles are similar to each other and we also should know that  in similar triangles, if the sides are in the ratio  \frac{m}{n} , the areas of the triangles are in the ratio  (\frac{m}{n})^2 . 
 
For example, say we have two similar triangles having sides {3, 4, 5} and {6, 8, 10}. The ratio of their corresponding sides is  \frac{3}{6}=\frac{1}{2} , so the areas of the triangles are in the ratio  (\frac{1}{2})^2=\frac{1}{4} . Let's check: the area of the first triangle is  \frac{3*4}{2}=6  and the area of the second triangle is  \frac{6*8}{2}=24 . The ratio of the areas  =\frac{6}{24}=\frac{1}{4} 
 
So, according to the above we can deduce that since triangles M and N are similar and the ratio of their sides is 3, then the ratio of their areas will be  3^2=9  (the area of triangle M is 9 times the area of triangle N).
 
(1) The ratio of the area of equilateral triangle M to the area of equilateral triangle N is 9:1
 
Well, first of all we knew that from the stem. Also, we need to find the perimeter of triangle N and so far we don't know the length of anything, just ratios. Not sufficient.
 
(2) The sum of the circumferences of circles circumscribing triangles M and N is  12π 
 
The radius of a circle circumscribing an equilateral triangle is  R=a*\frac{\sqrt{3}}{3} , where  a  is the length of a side of the triangle. So, if the length of a side of triangle N is  n  and the length of a side of triangle M is  m=3n , then this statement says that:  2\pi R_M+2\pi R_N=2\pi(3n*\frac{\sqrt{3}}{3})+2\pi(n*\frac{\sqrt{3}}{3})=12\pi :
 
 2\pi(3n*\frac{\sqrt{3}}{3})+2\pi(n*\frac{\sqrt{3}}{3})=12\pi 
 
 8(n*\frac{\sqrt{3}}{3})=12 
 
 n=\frac{3\sqrt{3}}{2} . We know the length of a side of triangle N, so the perimeter will be thrice that. Sufficient.
 
Notice that we could omit all above calculations and could have solved this statement even not knowing  R=a*\frac{\sqrt{3}}{3} . The important thing is to realize that we could get  R_N  and  R_M  (the radii of circumscribing circles) in terms of  n  (the side of an equilateral triangle N) and this get  2\pi(3n*some \ ratio)+2\pi(n*\ the \ same \ ratio)=12\pi . Here,  \pi  will be reduced and we are left with linear equation with one unknown,  n . We can solve and thus get the perimeter.

Answer: B

Raxit85 · Answer

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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Snezanelle · Answer

Don't we have this information in the description of the problem?

Raxit85 · Answer

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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chetan2u · Answer

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

B

Raxit85 · Answer

Hi, Chetan2u, can you please elaborate statement 2 with diagram?

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chetan2u · Answer

Attached above

Rakesh1987 · Answer

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}} Please hit +1 Kudos if you liked the answer.

exc4libur · Answer

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/√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/√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)

GMATGuruNY · Answer

Statement 1:
Given similar triangles with corresponding sides in a ratio of x:y, the ratio of the areas = x²:y².
The prompt indicates that triangles M and N have corresponding sides in a ratio of 3:1.
Thus, the ratio of the areas = 3²:1² = 9:1.
Implication:
Statement 1 merely confirms the information given in the prompt and thus offers no new information.
INSUFFICIENT.

Statement 2:
 https://i.ibb.co/w73DxVn/M-and-N-inscribed.png

Let R = the radius in triangle M and r = the radius in triangle N.
Since the sum of the circumferences = 12π, we get:
2πR + 2πr = 12π
2π(R + r) = 12π
R + r = 6

Since M and N have corresponding sides in a ratio of 3:1, the two corresponding radii must also be in this ratio.
Thus:
R = 3r

Since we have two variables (R and r) and two distinct linear equations (R+r=6 and R=3r), we can solve for variables R and r.
Given the value of r, the size of triangle N will be FIXED, allowing us to determine the triangle's perimeter.
SUFFICIENT.

B

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

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