Rectangle DEFG is inscribed in an equilateral triangle. . .

Question

https://e-gmat.com/blogs/wp-content/uploads/2016/11/02_01-e1478606992239.png Rectangle DEFG is inscribed in an equilateral triangle as shown above. What is the area of triangle ABC? (1) The area of rectangle DEFG is 8√3 square units (2) The area of triangle AGD is 2√3 square units To read all our articles: Must Read articles to reach Q51 To solve question of the week: Question of the Week Take a stab at this fresh question from e-GMAT. Post your analysis below. Official Solution to be provided after receiving some good analyses.

https://e-gmat.com/blogs/wp-content/uploads/2016/12/Geometry-Ability-Quiz-4-e1480581478957.png

Indraksh · Answer

Is the answer option c ie both 1 and 2 are required

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rakaisraka · Answer

St 1 - Gives us area of rect but there can be n number of rectangles so we cant fnd any side of triangle - NS
St2 - Again it gives us the area of small triangles. With this we can find one side of rectangle and sides of small triangles but we can have diff sizes of such triangles.
Combining we can find sides of square , then sides of small triangles. With these we can find the side of triangle- so Suff.

acegmat123 · Answer

Area of equilateral triangle =sqrt(3)*AB^2 / 4

So AB=?

Statement 1:

DG*GF=8*sqrt(3)

Insufficient

Statement 2:

1/2*AG*DG=2*sqrt(3)

We know ADG is a 30-60-90 triangle

So AG*AG*sqrt(3)=4*sqrt(3)

AG=2

DG=2*sqrt(3)

Insufficient

Statement 1&2:

2*sqrt(3)*GF=8*sqrt(3)

GF=4

We know that

BF=AG=2

Therefore AB= 2+2+4=8

Area= sqrt(3)*8^2 /4 = 16*sqrt(3)

Sufficient

C

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rohit_arora · Answer

Statement1: insufficient because we can have multiple values of sides of rectangle DEFG.
Statement2: Insufficient again.
Combining:
Draw a perpendicular CP that intersects DE at O and AB at P.
We can also see that area of triangle ADG is 1/4 the area of DEFG.
Triangles AGD and PGD can be proved congruent.
DE parallel to AB.
DE=1/2AB (AG=GP=PF=FB, DE. DE=GP+PF, AB=AG+GP+PF+FB)
by mid point theorum,points D and E are mid points of AC and BC respectively.
We can easily prove Triangles ADO and ADG congruent.
so all triangles have equal area i.e.
Tri ADG,DGP,DOP,EOP,EPF,EFB,CDO,CEO.
(area)ABC=8*(area ADG)
Thanks. Post Kudos if you like.

EgmatQuantExpert · Answer

Solution

Hey,

Please find below the official solution.

• We are given that  ABC  is an  equilateral triangle . 
o This means, each angle of this triangle is  60^o .   
• Also,  DEFG  is a  rectangle .  
o Hence each angle of this rectangle is  90^o .

• Since the opposite sides of a rectangle are parallel,  DE || GF , and therefore,  DE || AB .  
o So,  ∠ CDE = ∠ CAB = 60^o  (corresponding angles)
o Similarly,  ∠ CED = ∠ CBA = 60^o .

https://e-gmat.com/blogs/wp-content/uploads/2016/12/T1-e1481910569920.png

• Now, let’s assume that side  GF = x  units and side  GD = y  units.

So, expressing all the unknown lengths in terms of  x and y , we get:

https://e-gmat.com/blogs/wp-content/uploads/2016/12/T2-e1481910547933.png

Note:  
  
• Since  Triangle  AGD  is a  30^o-60^o-90^o  triangle , the ratio of the sides are  AG : DG : AD = 1 : √3 : 2 
• Thus if  DG = y , then  AG = y/√3  and  AD = 2y/√3 
• And we can conclude the same for the sides of triangle  EFB .

From the above diagram, we can infer that –
 
• The Sides of an equilateral triangle  ABC = x+2y/√3

Thus, in order to find the area of the triangle, we need to know the  value of x and y .

Analyze Statement 1 independently

• The area of rectangle  DEFG  is  8√3  square units
 o That is,  xy = 8√3  . . . (1)
o Multiple values of  x and y  will satisfy this equation.

Therefore statement 1 is not sufficient to arrive at a unique answer.

Analyze Statement 2 independently

• The area of triangle  AGD  is  2√3  square units

Thus, we can write -  
•  1/2*AG*GD=2√3 
•  1/2*y/√(3 )*y=2√3 
•  y^2=2^2×3 
•  y=2√3  . . . (2)
• But we don’t know the value of  x  yet.

Therefore, statement 2 is not sufficient to arrive at a unique answer.

Analyze by combining statement 1 and 2

• Put the value of  y  from equation (2) in equation (1) and we will get:   x = 4  
• Since we now know the value of   x and y  , we can find the area of triangle.

Therefore both statement 1 and 2 are required to answer the question.

Hence the correct Answer is C

Thanks,
Saquib 
Quant Expert
e-GMAT

AnkithaSrinivas · Answer

can we solve using symmetry of the triangle? then option would be sufficient. 
--> x=2y

chetan2u , please help

chetan2u · Answer

Of course we can use symmetry to solve questions => if ratio of sides is a:b, then ratio of area becomes a^2:b^2

But it is not clear what do x and y stand for.

bumpbot · Answer

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Rectangle DEFG is inscribed in an equilateral triangle. . .

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