There is a string 15 units in length. By cutting the string, Thomas w

Question

There is a string  15  units in length. By cutting the string, Thomas wants to make two equilateral triangles. If the ratio of the areas of two triangles is  2:3 , what is the side length of the smaller triangle?

A.  2 
 
B.  3 
 
C.  5(\sqrt{6}-2)  
 
D.  3(\sqrt{5}-2)  
 
E.  2\sqrt{6}

GMATinsight · Accepted Answer

Ratio of Areas = Square of ratio of sides

i.e. Ratio of sides = √2/√3

i.e. Perimeters of two triangles = (Side1+Side2)*3 = (√2x+√3x)*3 = 15

i.e. x = 5/(√2+√3) = 5(√3-√2)

Smaller triangles's side = √2 x = √2*5(√3-√2) = 5(√6-2)

Answer: Option C

Bunuel : This thread to be moved into PS forum

Bunuel · Answer

_________________
Done. Thank you for noticing.

Kritisood · Answer

Hi  GMATinsight  is this a general property for all equilateral triangles? "Ratio of Areas = Square of ratio of sides"

GMATinsight · Answer

Kritisood Yes, this property is generalized for any set of similar figures (triangles, quadrilaterals, circles etc.)

ABCDEFU · Answer

i.e. x = 5/(√2+√3) = 5(√3-√2)

Hi,

How did you get the denominator to the numerator here?

MathRevolution · Answer

=>

Assume  x  and  y  are the side lengths of the two triangles where  x < y.

Then we have  3x + 3y = 15  or  x + y = 5.

Since the ratio of the areas of those triangles is  2:3,  we have  x^2 : y^2 = 2 : 3  or  x : y = √2 : √3.

Cross multiplying gives us  √3x = √2y, y = \frac{(√3x)}{√2} , and  y = (\frac{√6}{2})x  (by multiplying the numerator and denominator by  √2 ).

Substituting  y = (\frac{√6}{2})x  into  x + y = 5  gives us:

x + y = 5

x + (\frac{√6}{2})x = 5

(\frac{2}{2})x + (\frac{√6}{2})x = 5

\frac{√6+2}{2}x = 5.

Then  x = \frac{10}{√6+2}=\frac{10(√6-2)}{2}=5(√6-2).

Therefore, C is the answer.
Answer: C

jcerdae · Answer

Hi,
What is the mathematical operation you did in order to change from

i.e. Perimeters of two triangles = (Side1+Side2)*3 = (√2 x +√3 x )*3 = 15 
to 
i.e.  x  = 5/(√2+√3) = 5(√3-√2) ???
 
Thank you!!

Kritisood · Answer

Hi, they took x as common from (√2x+√3x)*3 = 15 
x(√2+√3)*3=15
x=5/(√2+√3) = 5(√3-√2)

bumpbot · Answer

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There is a string 15 units in length. By cutting the string, Thomas w

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