If The Perimeter of Quadrilateral ABCD is 4, What Is Its Area? GMAT Data Sufficiency

Question: If the perimeter of quadrilateral ABCD is 4, what is its area?

(1) All four sides of ABCD are of equal length.
(2) All four internal angles of ABCD are right angles.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.        
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.        
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.        
4. EACH statement ALONE is sufficient.        
5. Statements (1) and (2) TOGETHER are not sufficient.

“If the perimeter of quadrilateral ABCD is 4, what is its area?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation

Approach Solution 1:

It is given the perimeter of quadrilateral ABCD is 4,

Looking at statement 1 it says
All four sides of ABCD are of equal length.
=> ABCD can be a square or rhombus.
Having: ABCD's perimeter is four
=> 4/4 = 1 because AB = BC = CD = DA.
=> no diagonals found
=> insufficient
Looking at statement 2 it says
All four internal angles of ABCD are right angles.
This means ABCD is square or rectangle
=> Insufficient
Combining
(1) + (2) => ABCD is square
=> Area of ABCD = 2*2= 4

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.  

Correct Answer: C

Approach Solution 2:

There is another approach to answering this question which is Pretty simple
It is given the perimeter of quadrilateral ABCD is 4,
Statement 1 says - All four sides of ABCD are of equal length
The figure can be a rhombus with an area that is not equal to one or a square with an area that is equal to one therefore it is Insufficient
Statement 2 says - All four internal angles of ABCD are right angles
The figure could be a rectangle or a square therefore it is Insufficient
combining the two statement
The figure is a square with area because all of its sides and angles are equal making it Sufficient

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Correct Answer: C

Approach Solution 3:

(1) All four sides of ABCD are of equal length.
=> ABCD can be a rhombus or square
Having: Perimeter of ABCD is 4 => AB = BC = CD = DA = 4/4 = 1.
=> Can't find the diagonals
=> Insufficient
(2) All four internal angles of ABCD are right angles.
=> ABCD is square or rectangle
=> Insufficient
(1) + (2) => ABCD is square => Area of ABCD = 2*2= 4=> 

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Correct Answer: C

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