If a triangle ABC is there such that BE=EF=FC=3 and DB=GC=2BE, what is

Question

If a triangle ABC is there such that BE=EF=FC=3 and DB=GC=2BE, what is the perimeter of a pentagon ADEFG?
 A. 9+6√3 
B. 6+4√3 
C. 9+6√2 
D. 6+4√2 
E. 4+2√3

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nalinnair · Answer

I could solve this question after assuming that DE and GF are perpendicular to side BC, making Triangles DBE and GFC right triangles.
Since we know BE = 3 and BD = 6, we can calculate DE. DE =  3\sqrt{3} . Similarly, GF =  3\sqrt{3} .
Now, the sides of Triangles DBE and GFC are in a ratio of  1:\sqrt{3}:2 . Hence, these are 30-60-90 triangles with angle DBE = angle GCF = 60. 
This implies that angle BAC =60 and Triangle ABC is an equilateral triangle. AB = BC = AC = 9
AD = AB - DB = 9 - 6 = 3
Similarly, AG = AC - GC = 9 - 6 = 3
So, perimeter of pentagon ADEFG = AD + AG + DE + GF + EF =  3 + 3 + 3\sqrt{3} + 3\sqrt{3} + 3 = 9 + 6\sqrt{3} 
Hence, the correct answer is A.

However, the above solution is dependent on the assumption that DE and GF are perpendicular to BC. Probably, the question should provide this information either in the question stem or in the figure. 
Is there any way we can conclude that DE and GF are perpendicular to BC based on the given information ? Or are there any methods for calculating the perimeter without relying on the fact that DE and GF are perpendicular to BC?
 chetan2u  - Can you help?

MathRevolution · Answer

Since BE=2 and DB=4, degree B = degree C=60 degree. So, DE=GF=3√3 and AD=AG=3. Hence, the perimeter of a pentagon ADEFG is 3+3+3+3√3+3√3=9+6√3.Hence,the correct answer is A.

FlyingCycle · Answer

Hi  MathRevolution ,

how did you get this "Since BE=2 and DB=4, degree B = degree C=60 degree."

Question has given BE=3 right?

stne · Answer

FlyingCycle ,

You have a valid point, that is a typo. It is already given BE=3 hence it should have been BE=3 and DB=6.

In a 30 : 60 : 90 triangle sides are in the ratio  x : x\sqrt3 : 2x 
From the figure we see ED is perpendicular to BC hence angle DEB = 90

Now using the given information that two sides are in the ratio  x  and  2x  and one angle is 90 we can conclude it's a 30 60 90  \Delta . Hence angle B ( and also angle C) is 60.

Hope it's clear.

Bunuel ,  MathRevolution , Sir please can you correct the typo, so as to prevent others too from getting confused. Thank you.

jrk23 · Answer

In question its not mentioned its a right triangle.... just by seeing the picture can we say its a right angle and assume angle c is 60 degree?

stne · Answer

Someone mentioned similar doubt. I don't think question can be solved without such assumption.

Posted from my mobile device

FlyingCycle · Answer

I don't think in GMAT we would get such ambiguity in the question. We cannot just assume whatever we want and get the answer

stne point is also right that without assumption we wouldn't get the answer

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If a triangle ABC is there such that BE=EF=FC=3 and DB=GC=2BE, what is

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