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Rectangle DEFG is inscribed in an equilateral triangle. . .

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EgmatQuantExpert
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Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   Updated on: 07 Aug 2018, 06:59
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A
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C
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E

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65% (hard)

Question Stats:

59% (02:17) correct 41% (02:31) wrong based on 286 sessions
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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

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Official Answer
C

Originally posted by EgmatQuantExpert on 08 Nov 2016, 05:13.
Last edited by EgmatQuantExpert on 07 Aug 2018, 06:59, edited 2 times in total.
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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   16 Dec 2016, 11:08
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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, \(∠ CED = ∠ CBA = 60^o\).



    • 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:



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


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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   08 Nov 2016, 11:34
Is the answer option c ie both 1 and 2 are required

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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   13 Nov 2016, 05:46
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.
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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   13 Nov 2016, 07:28
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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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   13 Nov 2016, 10:16
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.
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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   13 Jun 2021, 22:27
can we solve using symmetry of the triangle? then option would be sufficient.
--> x=2y

chetan2u, please help
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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   14 Jun 2021, 22:21
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AnkithaSrinivas wrote:
can we solve using symmetry of the triangle? then option would be sufficient.
--> x=2y

chetan2u, please help



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.
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Re: Rectangle DEFG is inscribed in an equilateral triangle. . . [#permalink] New post   15 Jul 2022, 07:23
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