Rectangle R has an area A square units. Is the rectangle R a square?

Question

(1) A = 36 square units, and sides of rectangle R are in integer units.

(2) For all the given rectangles with area A, R has minimum perimeter.

Princ · Answer

OA: B

Rectangle R has an area A square units. Is the rectangle R a square?

Statement 1: A = 36 square units, and sides of rectangle R are in integer units.
 Possible combinations of dimensions of side are (1,36);(2,18);(3,12);(4,9);(6,6)
 Only one combination results into square.
 So statement 1 is insufficient for answering :Is the rectangle R a square?

Statement 2 :For all the given rectangles with area A, R has minimum perimeter.
 Let the length and width of rectangle be x and y.
 Perimeter(P)= 2*(x+y)  ; Area(A)=  x*y 
 taking arithmetic mean and geometric mean of both sides x and y , we get 
 A.M =  \frac{x+y}{2}  ; G.M =  \sqrt{x*y}   
 as we know  arithmetic mean\geq{geometric mean} 
 \frac{x+y}{2}\geq{\sqrt{x*y}} 
Multiplying both side by 4 , we get 
 2*(x+y)\geq{4*\sqrt{x*y}} 
 P\geq{4*\sqrt{A}} 
So Lowest value of Perimter of rectangle is  4*\sqrt{A} 
It occurs when Arithmetic mean and Geometric mean of the sides of rectangle are equal.
A.M= G.M occurs only when both the term are equal. In this case, it will occurs when x=y.
As it is given in question stem that R is rectangle, and statement 2 is implying that length = width for rectangle R.
So Rectangle R is a square.
Statement 2 is alone sufficient

push12345 · Answer

1) A = 36 square units, and sides of rectangle R are in integer units. 
Can be multiple options
4*9
6*6
12*3
36*1
Can be square or cannot be square.Not sufficient

(2) For all the given rectangles with area A, R has minimum perimeter

Square is a rectangle with min perimeter.
But rectangle is not a square

Eg area=36 ,let perimeter be P
4*9. P=13
6*6 P=12
2*18 P=20
1*36 P=37
3*12 P=15

So 6*6 has min area hence proved
Sufficient

B is answer

Give kudos if it helps

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Rectangle R has an area A square units. Is the rectangle R a square?

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GMAT Data Sufficiency (DS) Questions

Rectangle R has an area A square units. Is the rectangle R a square?

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