A Rhombus is said to be mystic quadrilateral if its diagonals are in

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GMATBusters’ Quant Quiz Question -5

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A Rhombus is said to be mystic quadrilateral if its diagonals are in the ratio  1:\sqrt{3} . Is the quadrilateral ABCD a mystic quadrilateral?

(1) Angle ABC = 60 deg
(2) Diagonals are perpendicular and  (\frac{diag1}{diag2})^2 = 3

GMATBusters · Answer

Official Solution:

A Rhombus is said to be mystic quadrilateral if its diagonals are in the ratio  1:\sqrt{3} . Is the quadrilateral ABCD a mystic quadrilateral?

(1) Angle ABC = 60 deg
A mystic quadrilateral is defined for a specific Rhombus. from Statement 1, it is not clear whether quadrilateral ABCD is a Rhombus or not?
NOT Sufficient.

(2) Diagonals are perpendicular and  (\frac{diag1}{diag2})^2 = 3
A mystic quadrilateral is defined for a specific Rhombus. from Statement 1, it is not clear whether quadrilateral ABCD is a Rhombus or not?
NOT Sufficient.

Even after combining Statements 1 & 2, we can not say that quadrilateral ABCD is a mystic quadrilateral or not?
Answer E

Raxit85 · Answer

A Rhombus is said to be mystic quadrilateral if its diagonals are in the ratio 1:root3. Is the quadrilateral ABCD a mystic quadrilateral?
I don't know the exact approach, still i'll give a try.

Diagonal 1 of quadrilateral/diagonal 2 of quadrilateral = 1/root3 = 1/1.73

1) Angle ABC = 60 deg 
<B = 60 so, <A + <C= 120. One of the diagonal is root3 means that triangle involves 30-60-90. 
Not sufficient.

2) Diagonals are perpendicular and (diag1/diag2)2= 3
(d1/d2)2 = 3/1
d1/d2 = root3/1.
I think it's sufficient condition to be mystic quadrilateral as per definition. So sufficient.

Ans. B

Archit3110 · Answer

d1/d2 = 1/√3
#1
angle ABC = 60*
no relation of diagonals insufficient
#2
 Diagonals are perpendicular and (diag1diag2)2= 3
(d1/d1)^2 = 3
square both sides
d1/d2 = √3/1
but we dont know about side values
insufficient
from 1 &2
we know that we get 4 ; 30:60:90 ∆ with sides 1/2: √3/2 : 1 
knowing this we can say that length of the side of diagonal 1;√3 ; 
sufficient
OPTION C

A Rhombus is said to be mystic quadrilateral if its diagonals are in the ratio 1:root3. Is the quadrilateral ABCD a mystic quadrilateral?
1) Angle ABC = 60 deg
2) Diagonals are perpendicular and (diag1diag2)2(diag1diag2)2= 3

uc26 · Answer

We need to first figure out if quadrilateral ABCD is a rhombus and if yes, we need to check if the diagonals are in the ratio 1: root 3

1) Not enough information to figure out if ABCD is a rhombus. Also, no information on the diagonals. Not sufficient.
2) ABCD can be a kite or a rhombus. Not sufficient.
1&2) No new information to figure out if ABCD is a rhombus even though we have the ratio of the square of the diagonals of ABCD. We also do not have any information on the sides of the ABCD. For it to be a rhombus, all the sides must be equal.
ABCD can be a kite or a rhombus. Not sufficient.
Answer: E

ashwini2k6jha · Answer

find d1/d2=1/root3?
state1, not sufficient
angle ABC=60 NOTHING can be said about d1 and d2
state2,
diagonal perpendicular,it's a rhombus
d1/d2=1/root3
hence sufficient
Answer B

1BlackDiamond1 · Answer

Answer is B.

Statement 1 is not sufficient to answer the question.
Statement 2 is sufficient to answer the question.

freedom128 · Answer

A rhombus is a parallelogram which opposite angles have equal measure.

Q. Is the quadrilateral ABCD a mystic quadrilateral?

(1) Angle ABC = 60 deg 
It is not clear whether quadrilateral ABCD is a rhombus or not. 
NOT SUFFICIENT

(2) Diagonals are perpendicular and (diag1 /diag2)^2 = 3 
It is not clear whether quadrilateral ABCD is a rhombus or not. It can be a kite. 
NOT SUFFICIENT

Combined 
It is still not clear whether quadrilateral ABCD is a rhombus or not. It can be a kite. 
NOT SUFFICIENT

FINAL ANSWER IS (E)

Posted from my mobile device

shameekv1989 · Answer

ABCD quadrilateral can be a Kite-shaped in which diagonals are perpendicular to each other with ratio of \(\sqrt{3}:1\)and angle ABC = 60 degrees. We need to know whether the diagonals are angle bisectors or not to come to a conclusion whether the quadrilateral is a rhombus.

Answer - E

Answer - E

StevenA · Answer

A Rhombus is said to be mystic quadrilateral if its diagonals are in the ratio 1:root3. Is the quadrilateral ABCD a mystic quadrilateral?
1) Angle ABC = 60 deg
2) Diagonals are perpendicular and (diag1diag2)2(diag1diag2)2= 3

If angle ABC is 60 deg, angle BCA is 120 deg and angle CDA is 60 degree. Hence, triangle ABC is an equilateral triangle and all length are the same (let's say X)
Diagonal AC = AB = BC = X
Now, the longer diagonal AB's length needs to be found. Use triangle ACB which all the angles are found. From there, AB=  \sqrt{3} 
hence, the ratio is 1: root 3

for Statement 2, since (diag 1 / diag2)^2 is 3, they will be in the ratio 1: root 3

Hence, both statements individually are sufficient to answer the question. D

Kinshook · Answer

Given: A Rhombus is said to be mystic quadrilateral if its diagonals are in the ratio 1:root3.
Asked: Is the quadrilateral ABCD a mystic quadrilateral?

1) Angle ABC = 60 deg
Case 1: Quadrilateral ABCD is NOT a rhombus - > It is also not a mystic quadrilateral.
Case 2: Quadrilateral ABCD is a rhombus.
Let the side of the rhombus be x

Attachment:

Screenshot 2020-05-10 at 9.51.28 AM.png [ 21.02 KiB | Viewed 2557 times ]

\(DE = x/2\sqrt{3}\); BE = x + x/2 = 3x/2
\(BD = \sqrt{(DE^2 + BE^2)} = \sqrt{(3x^2/4 + 9x^2/4)}= x/2\sqrt{(3+9)}= x\sqrt{3}\)
\(AF = x\sqrt{3}\); FC = x/2
\(AC = \sqrt{(AF^2 + FC^2)} =\sqrt{(3x^2/4 + x^2/4)} = x/2\sqrt{(3+1)}= x\)
\(\frac{BD}{AC} = \sqrt{3}\)
If ABCD is a rhombus, it is a mystic quadrilateral
Since it is not provided whether ABCD is a rhombus.
NOT SUFFICIENT

2) Diagonals are perpendicular and (diag1/diag2)^2= 3
Diag1/diag2 = root3/1
A quadrilateral may or may not be a rhombus if diagonals are perpendicular. It becomes a parallelogram if diagonals are perpendicular and are also bisecting each other.
NOT SUFFICIENT

(1) + (2)
1) Angle ABC = 60 deg
2) Diagonals are perpendicular and (diag1/diag2)^2= 3
Since it is still not provided whether ABCD is a rhombus.
NOT SUFFICIENT

IMO E

Hi GMATBusters
Please check what is the flaw in the solution.

GMATBusters · Answer

The flaw in your solution is as follows: https://gmatclub.com/forum/download/file.php?id=54453 I have taken DE = 1 similarly, you can calculate other diagonal Also, I feel it can better be proved this way: https://gmatclub.com/forum/download/file.php?id=54452 if any doubt, kingly tag. CamScanner 05-11-2020 08.30.46_1.jpg CamScanner 05-11-2020 08.30.46_2.jpg

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A Rhombus is said to be mystic quadrilateral if its diagonals are in

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