Jeff is painting two murals on the front of an old apartment

Question

https://gmatclub.com/forum/download/file.php?id=23098 Jeff is painting two murals on the front of an old apartment building that he is renovating. One mural will cover the quadrilateral face ABCD while the other will cover the circular face (shown to the right, with radius XY). Assuming that the thickness of the coats of paint is negligible, will each mural require the same amount of paint? Note: Figures are not drawn to scale. (1) AB = BC = CD = DA and AB=XY\sqrt{\pi} (2) AC = BD and AC=XY\sqrt{2\pi} Untitled.png

giddi77 · Accepted Answer

Is it C?

1. ABCD could be square or a rhombus. 
If ABCD is a square, then area of Circle = area of ABCD
If ABCD is a rhombus with sat 60-120-60-120 degress,
 Area of ABCD < AB^2 = XY^*pi
INSUFF

2. AC = BD = XY*SQRT(2pi).
ABCD could be a rectangle or a square or a rhombus.
IF ABCD is a rectangle, Area of ABCD is smaller than the area of the Circle
If ABCD is a square, both areas are equal.
Rhombus.. unequal.

Combine (1) & (2), ABCD has to be a square with the same area as that of the circle.

giddi77 · Answer

chiragr-buddy, I think there is no light to show here.. There was a blackout in Manhattan when this question was created and explained by 99%tile students

They probably didn't know that a regular trapezium also has equal diagonals. If the diagonals are equal and also bisect each other, then the corners have to be 90 degrees. Still it's a very good question. We should remember that Sides equal => the quadrilateral could be a nice Square or an ugly Rhombus!

sm176811 · Answer

I wud Say A

From I we know that the quad is a square.

Hence the Area = AB^2 == Area of the circle (pi*XY^2).

From B. You could have multiple quadirlaterals!

If the two diagnals are perpendicular, then the given equation could be used to determine (the quad will then be a square or a kite). But since nothing is provided abt their angle of intersection, we cannot assume that.

Hence, A remains!

Whats the OA?

chiragr · Answer

it's quadirlaterals not square... it can be rhombus!

chiragr · Answer

for 2 AC=BD can not be rhombus, rhombus has equal sides not diagonal!

sm176811 · Answer

Agreed... It could be Rhombus or a square..

Hence C!

sm176811 · Answer

B could also generate a trapezoid!

giddi77 · Answer

Oops... take out rhombus. Still it can be a rectangle or square. Hence B is INSUFF?

chiragr · Answer

C is the correct answer, the only reason I have this question is following explaination

"
Statement 2 tells us that the diagonals are equal--thus telling us that ABCD has right angle corners (The only way for a quadrilateral to have equal diagonals is if its corners are 90 degrees.) Statement 2 also gives us a numerical relationship between the diagonal of ABCD and the radius of the circle. If we assume that ABCD is a square, this relationship would allow us to determine that the area of the square and the area of the circle are equal. However, once again, we cannot assume that ABCD is a square."

The only way for a quadrilateral to have equal diagonals is if its corners are 90 degrees.

I can not really visualize this! Can any one show me the light?

chiragr · Answer

now I can sleep without nightmares

TheMechanic · Answer

If the diagonals are equal it cant be a rhombus. It would be either a square or a rectangle.

Combining 1 and 2, we get C.

anairamitch1804 · Answer

St1 - ABCD is a rhombus. If it were a square, the condition would have implied paint is equal coz area would be equal. However, we dont know that. All we know is its a rhombus, and the area of the rhombus could be anything relative to the circle depending on the ratio of its diagonal to its side -> INSUFF

St2 - ABCD is a ||m. Again, a parallelogram has area = base x height. But we only know info about diagonals. In other words, for the same length of diagonals, you could have infinitely many areas of ||ms - so not enough info - INSUFF

Both combined - we know sides of ABCD are equal and so are the diagonals. Ratio of diagonal to side is sqrt(2) -> This means that ABCD is now a square. In this case area = piXY^2 = area of circle

Thus answer is C

kornn · Answer

Dear  Bunuel   VeritasKarishma   GMATGuruNY   EMPOWERgmatRichC ,

What are the possible quadrilaterals according to Statement 2?

Are these the possibilities for Statement 2?
1. rectangle 2. isosceles trapezoid 3. kite 4. irregular quadrilateral 5. square

If possible, could you provide the full solution?

KarishmaB · Answer

With 2 equal diagonals, you can make a whole lot of different quadrilaterals. The most generic one will be an equidiagonal quadrilateral and the most specific will be a square. Read about equidiagonal quads here: https://en.wikipedia.org/wiki/Equidiago ... drilateral The square will have same area as circle while other figures will not so this statement alone will not be sufficient.

preetamsaha · Answer

ST(1) : AB = BC = CD = DA and AB = XY√π
CASE I : let ,ABCD is a square, area of ABCD =(AB)^2 , area of Circle = π(XY)^2 =(AB)^2 ; =>EQUAL
CASE II : let ,ABCD is a rhombus ,
since area of rhombus < area of an square of same side
Area of ABCD ≠ area of Circle
so, ST (1) insufficient .

ST(2) : AC = BD and AC = XY√2π
AC =BD i.e. diagonals are equal .
CASE I : let ,ABCD is a square, area of ABCD =(AB)^2 =(AC/√2)^2=(AC)^2/2, area of Circle = π(XY)^2 =(AC)^2/2 ; =>EQUAL
CASE II : let ,ABCD is a rectangle ,
since area of rectangle < area of an square of same diagonal [ suppose , d= 5 , area of square = (5/√2)^2= 12.5 whereas area of rectangle = 3*4 = 12 ( 5^2 = 3^2+4^2) ]
Area of ABCD ≠ area of Circle
so, ST (2) insufficient .

Combine (1) & (2), ABCD has to be a square ( all sides are equal and diagonals are equal )
Area of ABCD = area of Circle. [b] sufficient[/b]

correct answer is C

SreyaSanghai · Answer

Can a question of this difficulty level appear in GMAT?

JeffTargetTestPrep · Answer

We need to answer the question:

Is area_ABCD = pi(XY)^2 ?

Statement One Alone:

=> AB = BC = CD = DA and AB = XY√pi

Squaring the second equation, we have:

AB^2 = pi(XY)^2

Using the above information, we can rephrase the question:

Is area_ABCD = AB^2 ?

Since AB = BC = CD = DA, the sides of the quadrilateral are equal.

If quadrilateral ABCD is a square, then the answer is Yes. Whereas, if ABCD is a rhombus but not a square, then the answer is No since area_ABCD would be less than AB^2.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> AC = BD and AC = XY√(2pi)

From the second equation, we have:

AC^2 = 2pi(XY)^2

(AC^2)/2 = pi(XY)^2

Using the above information, we can rephrase the question:

Is area_ABCD = (AC^2)/2 ?

Since AC = BD, the diagonals of the quadrilateral are equal.

If quadrilateral ABCD is a square, then the answer is Yes since a square’s area can be determined as (diagonal^2)/2. Whereas, if ABCD is a rectangle but not a square, then the answer is No since area_ABCD would be less than (AC^2)/2.

Statement two is not sufficient. Eliminate answer choice B.

Statements One And Two Together:

Since the sides of the quadrilateral are equal and the diagonals of the quadrilateral are equal, quadrilateral ABCD must be a square. In this case, we have a definite Yes answer to the rephrased questions.

The two statements together are sufficient.

Answer: C

Jeff is painting two murals on the front of an old apartment

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Jeff is painting two murals on the front of an old apartment

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