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655-705 (Hard)|   Geometry|      
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mikemcgarry
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ABCD is a square, and EFGH is a square, each vertex of which is on a s 27 May 2016, 17:05
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55% (hard)

Question Stats:

59% (02:10) correct 41% (02:44) wrong based on 145 sessions

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Square inside square.png
Square inside square.png [ 4.88 KiB | Viewed 7754 times ]
ABCD is a square, and EFGH is a square, each vertex of which is on a side of ABCD. What is the ratio of the area of square EFGH to the area of square ABCD?

Statement #1: AE:AB = 4:7

Statement #2: The ratio of the area of triangle AHE to the area of square EFGH is 0.24

Geometry is a truly beautiful subject! This question is one of a set of ten practice DS questions about geometry. To see the others, as well as the OE for this particular question, see:
GMAT Data Sufficiency Geometry Practice Questions

Mike :-)
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chetan2u
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GMAT Focus 1: 735 Q90 V89 DI81
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ABCD is a square, and EFGH is a square, each vertex of which is on a s 27 May 2016, 19:53
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mikemcgarry
Attachment:
Square inside square.png
ABCD is a square, and EFGH is a square, each vertex of which is on a side of ABCD. What is the ratio of the area of square EFGH to the area of square ABCD?

Statement #1: AE:AB = 4:7

Statement #2: The ratio of the area of triangle AHE to the area of square EFGH is 0.24

Geometry is a truly beautiful subject! This question is one of a set of ten practice DS questions about geometry. To see the others, as well as the OE for this particular question, see:
GMAT Data Sufficiency Geometry Practice Questions

Mike :-)
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we have two squares...
the one inside, EFGH, will have least area when its vertex is in center of the sides.....
and will become max as it moves closer to the vertex of the bigger square ABCD..

Second point is that all the triangles formed on the vertex of ABCD will be similar...

lets see the statement

Statement #1: AE:AB = 4:7
let the common ratio be x....
so AE = 4x and AB = 7x, so BE = 3x...

side of ABCD = 7x and side of EFGH = HYP of triangle whose sides are 3x and 4x = \(\sqrt{(3x)^2+(4x)^2}\)..
ratio =\(\frac{7x*7x}{\sqrt{(3x)^2+(4x)^2}^2}\)..
variable x will get cancelled out and we will have a numeric value...
Suff

Statement #2: The ratio of the area of triangle AHE to the area of square EFGH is 0.24
all triangles are similar so their Area = 0.24*Area of EFGH*4.....
Let area of square EFGH = x
Area of ABCD = Area of 4 triangles + area of EFGH = 0.24*x*4 + x...

ratio =\(\frac{0.24*x*4 + x}{x}\) = 1.96/1
Suff

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Ashishsteag
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ABCD is a square, and EFGH is a square, each vertex of which is on a s 05 Jul 2016, 03:13
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chetan2u
mikemcgarry
Attachment:
Square inside square.png
ABCD is a square, and EFGH is a square, each vertex of which is on a side of ABCD. What is the ratio of the area of square EFGH to the area of square ABCD?

Statement #1: AE:AB = 4:7

Statement #2: The ratio of the area of triangle AHE to the area of square EFGH is 0.24

Geometry is a truly beautiful subject! This question is one of a set of ten practice DS questions about geometry. To see the others, as well as the OE for this particular question, see:
GMAT Data Sufficiency Geometry Practice Questions

Mike :-)

we have two squares...
the one inside, EFGH, will have least area when its vertex is in center of the sides.....
and will become max as it moves closer to the vertex of the bigger square ABCD..

Second point is that all the triangles formed on the vertex of ABCD will be similar...

lets see the statement

Statement #1: AE:AB = 4:7
let the common ratio be x....
so AE = 4x and AB = 7x, so BE = 3x...

side of ABCD = 7x and side of EFGH = HYP of triangle whose sides are 3x and 4x = \(\sqrt{(3x)^2+(4x)^2}\)..
ratio =\(\frac{7x*7x}{\sqrt{(3x)^2+(4x)^2}^2}\)..
variable x will get cancelled out and we will have a numeric value...
Suff

Statement #2: The ratio of the area of triangle AHE to the area of square EFGH is 0.24
all triangles are similar so their Area = 0.24*Area of EFGH*4.....
Let area of square EFGH = x
Area of ABCD = Area of 4 triangles + area of EFGH = 0.24*x*4 + x...

ratio =\(\frac{0.24*x*4 + x}{x}\) = 1.96/1
Suff

D
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Dear sir,

I don't get the first part...
AE/AB=4/7
AE/AE+EB=4/7
AE/EB=4/3

So,AE=4x and EB=3x
But then how EBF or any other triangle is a 3-4-5 triangle?? :?
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mikemcgarry
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ABCD is a square, and EFGH is a square, each vertex of which is on a s 05 Jul 2016, 10:31
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Ashishsteag
Dear sir,

I don't get the first part...
AE/AB=4/7
AE/AE+EB=4/7
AE/EB=4/3

So,AE=4x and EB=3x
But then how EBF or any other triangle is a 3-4-5 triangle?? :?
Show more
Dear Ashishsteag

This is Mike McGarry, the author of this question. I'm happy to respond. :-)

So I believe you get that, for some unknown x, AE = 4x and EB = 3x.

You see, the entire figure has four-fold symmetry. We could rotate the entire shape by 90 degrees and it would be the same. The four triangles, {AEH, BFE, CGF, and DHG} have to be entirely congruent. Thus,

AE = BF = CG = DH = 4x
AH = BE = CF = GD = 3x

Thus, all of them are 3-4-5 triangles.

Does this make sense?
Mike :-)
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ABCD is a square, and EFGH is a square, each vertex of which is on a s 05 Jul 2016, 22:30
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mikemcgarry
Ashishsteag
Dear sir,

I don't get the first part...
AE/AB=4/7
AE/AE+EB=4/7
AE/EB=4/3

So,AE=4x and EB=3x
But then how EBF or any other triangle is a 3-4-5 triangle?? :?
Dear Ashishsteag

This is Mike McGarry, the author of this question. I'm happy to respond. :-)

So I believe you get that, for some unknown x, AE = 4x and EB = 3x.

You see, the entire figure has four-fold symmetry. We could rotate the entire shape by 90 degrees and it would be the same. The four triangles, {AEH, BFE, CGF, and DHG} have to be entirely congruent. Thus,

AE = BF = CG = DH = 4x
AH = BE = CF = GD = 3x

Thus, all of them are 3-4-5 triangles.

Does this make sense?
Mike :-)
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Thanx a lot mike....:D
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ABCD is a square, and EFGH is a square, each vertex of which is on a s 26 Apr 2019, 20:19
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mikemcgarry, I do not fully understand why 2nd statement is sufficient. Can you please help explain?
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