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In the figure above, the measure of angle EAB in triangle ABE is 90...

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In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.


Attachment:
ZZ1.jpg
ZZ1.jpg [ 12.5 KiB | Viewed 774 times ]

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Originally posted by CAMANISHPARMAR on 29 Jun 2018, 10:00.
Last edited by Bunuel on 29 Jun 2018, 10:05, edited 1 time in total.
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Re: In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 11:20
CAMANISHPARMAR wrote:
Image

In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.


Attachment:
ZZ1.jpg


Question stem:- AB=?
Statement1:-

Given \(\frac{0.5*AE *AB}{EB^2}=\frac{6}{25}\)
Let AB=x unit , hence \(x^2+12^2=EB^2\)
Substituting , we have, \(\frac{0.5*12*x}{x^2+12^2}=\frac{6}{25}\)
Or,\(x^2-25x+12^2=0\)
Since we have two positive roots which implies more than one value of AB. So, st1 is insufficient.

Statement2:-
Given,\(\frac{AE}{AB}=\frac{3}{4}\) & \(BE=\frac{perimeter}{4}=\frac{80}{4}=20\)
Now, we can definitely obtain the length of AB by using Pythagoras property in the right angled triangle AEB.(\((3k)^2+(4k)^2=20^2\), k can be calculated , so AB=4k)
Therefore, st2 is sufficient.

Ans. (B)
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In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 11:41
CAMANISHPARMAR wrote:
Image

In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.


Attachment:
ZZ1.jpg


Always look for SPECIAL TRIANGLES.

Statement 1:
Since the length of AE is divisible by 3 and 4, test whether it's possible that ABE is a multiple of a 3:4:5 triangle.

Case 1: AE=12, AB=16 and BE=20, with the result that AE:AB:BE = 3:4:5
In this case:
\(\frac{ABE}{BCDE} = \frac{(0.5 * 12 *16)}{20^2} = \frac{96}{400} = \frac{6}{25}\).

Case 2: AE=12, AB=9 and BE=15. with the result that AB:AE:BE = 3:4:5
In this case:
\(\frac{ABE}{BCDE} = \frac{(0.5 * 12 * 9)}{15^2} = \frac{54}{225} = \frac{6}{25}\).

Statement 1 is satisfied by both cases.
Since AB can be different values, INSUFFICIENT.

Statement 2:
Statement 2 is satisfied only by Case 1.
Thus, AB=16.
SUFFICIENT.


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Re: In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 11:52
GMATGuruNY wrote:
CAMANISHPARMAR wrote:
Image

In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.


Attachment:
ZZ1.jpg


Always look for SPECIAL TRIANGLES.

Statement 1:
Since the length of AE is divisible by 3 and 4, test whether it's possible that ABE is a multiple of a 3:4:5 triangle.

Case 1: AE=12, AB=16 and BE=20, with the result that AE:AB:BE = 3:4:5
In this case:
\(\frac{ABE}{BCDE} = \frac{(0.5 * 12 *16)}{20^2} = \frac{96}{400} = \frac{6}{25}\).

Case 2: AE=12, AB=9 and BE=15. with the result that AB:AE:BE = 3:4:5

In this case:
\(\frac{ABE}{BCDE} = \frac{(0.5 * 12 * 9)}{15^2} = \frac{54}{225} = \frac{6}{25}\).

Statement 1 is satisfied by both cases.
Since AB can be different values, INSUFFICIENT.

Statement 2:
Statement 2 is satisfied only by Case 1.
Thus, AB=16.
SUFFICIENT.



Hi,

How did u arrive at case 2, AB=9 unit?

Case-I is clearly understood(3x:4x:5x with x=4).
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Re: In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 12:11
PKN wrote:
Hi,

How did u arrive at case 2, AB=9 unit?

Case-I is clearly understood(3x:4x:5x with x=4).


Because AE=12 is a multiple of both 3 and 4, I tested whether it could constitute the SMALLEST side of a 3:4:5 triangle and whether it could constitute the MIDDLE side of a 3:4:5 triangle.

In Case 1, AE=12 constitutes the SMALLEST side of a 3:4:5 triangle:
AE --> 3*4 = 12
AB --> 4*4 = 16
BE --> 5*4 = 20

In Case 2, AE=12 constitutes the MIDDLE side of a 3:4:5 triangle:
AB = 3*3 = 9
AE = 4*3 = 12
BE = 5*3 = 15

As shown in my earlier solution, both cases satisfy Statement 1.
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In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 18:35
CAMANISHPARMAR wrote:
Image

In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.


Attachment:
ZZ1.jpg



Triangle ABE is right angled \(\triangle\) and its hypotenuse BE is also the side of square ABCD..

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}=6x/25x\).
area of \(\triangle {AEB}\)= \(\frac{1}{2}12*AB=6x.....AB=x\)
Area of square = \(25x=BE^2=AB^2+AE^2=12^2+x^2\)......
\(25x=x^2+144.......................x^2-25x+144=0.....................(x-9)(x-16)=0\)
so x= AB can be 16 or 9
insuff

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.
so BE = \(\frac{80}{4}=20\)....
ratio of AE:AB = 3:4 means it is 3:4:5
so AE:AB:BE=3x:4x:5x where 5x=20....x=4
so AB=4*4=16
suff

B
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In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 18:52
1
chetan2u wrote:
CAMANISHPARMAR wrote:
Image

In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.


Attachment:
ZZ1.jpg



Triangle ABE is right angled \(\triangle\) and its hypotenuse BE is also the side of square ABCD..

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).
area of \(\triangle {AEB}\)= \(\frac{1}{2}12*AB=6*\sqrt{BE^2-AB^2}\)
Area of square = BE^2
ratio = \(\frac{6*square_root(BE^2-AB^2)}{BE^2}=\frac{6}{25}\)
two unknowns and there is noway any variable will get cancelled out

insuff

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.
so BE = \(\frac{80}{4}=20\)....
ratio of AE:AB = 3:4 means it is 3:4:5
so AE:AB:BE=3x:4x:5x where 5x=20....x=4
so AB=4*4=16
suff

b



Hi chetan2u,

I think it is: AB=\(\sqrt{(BE^2-AE^2)}\)

And the ratio becomes an equation with one unknown variable.
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Re: In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 19:28
PKN wrote:
chetan2u wrote:
CAMANISHPARMAR wrote:
Image

In the figure above, the measure of angle EAB in triangle ABE is 90 degrees, and BCDE is a square. What is the length of AB?

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.


Attachment:
ZZ1.jpg



Triangle ABE is right angled \(\triangle\) and its hypotenuse BE is also the side of square ABCD..

(1) The length of AE is 12, and the ratio of the area of triangle ABE to the area of square BCDE is \(\frac{6}{25}\).
area of \(\triangle {AEB}\)= \(\frac{1}{2}12*AB=6*\sqrt{BE^2-AB^2}\)
Area of square = BE^2
ratio = \(\frac{6*square_root(BE^2-AB^2)}{BE^2}=\frac{6}{25}\)
two unknowns and there is noway any variable will get cancelled out

insuff

(2) The perimeter of square BCDE is 80 and the ratio of the length of AE to the length of AB is 3 to 4.
so BE = \(\frac{80}{4}=20\)....
ratio of AE:AB = 3:4 means it is 3:4:5
so AE:AB:BE=3x:4x:5x where 5x=20....x=4
so AB=4*4=16
suff

b



Hi chetan2u,

I think it is: AB=\(\sqrt{(BE^2-AE^2)}\)

And the ratio becomes an equation with one unknown variable.


Thanks...
typo.. Dont know what I was thinking
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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 20:51
x=AB
12x/2 : 144+x^2 = 6 : 25
since Hypotenuse = sqrt(144+x^2)

6x / (6x+144+x^2)= 6/31 should cancel 6s
6x=36/31x+864/31+6/31x^2
186x=36x+864+6x^2
0=6x^2-150x+864
0=x^2-25x+144
0=(x-9)(x-16)
x=9,16 Insufficient

(2) square side=20
AE:20=3:4
AE=15
x=remaining side of right triangle side
Sufficient
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Re: In the figure above, the measure of angle EAB in triangle ABE is 90...  [#permalink]

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New post 29 Jun 2018, 21:11
gmatzpractice wrote:
x=AB
12x/2 : 144+x^2 = 6 : 25
since Hypotenuse = sqrt(144+x^2)

6x / (6x+144+x^2)= 6/31 should cancel 6s
6x=36/31x+864/31+6/31x^2
186x=36x+864+6x^2
0=6x^2-150x+864
0=x^2-25x+144
0=(x-9)(x-16)
x=9,16 Insufficient

(2) square side=20
AE:20=3:4
AE=15
x=remaining side of right triangle side
Sufficient


Everything is correct but you are wrong in the coloured portion..
AE:AB=3:4
So it's 3:4:5 triangle and sides are AE:AB:EB=AE:AB:20 =3:4:5
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2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: In the figure above, the measure of angle EAB in triangle ABE is 90... &nbs [#permalink] 29 Jun 2018, 21:11
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