Is quadrilateral ABCD with sides w, x, y, and z a rectangle?

Question

https://gmatclub.com/forum/download/file.php?id=53528 Is quadrilateral ABCD with sides w, x, y, and z a rectangle? (1) w=x and z=y but w≠z (2) Perimeter of the quadrilateral is 2(x+y) Q.jpg

CareerGeek · Answer

Quadrilateral is a rectangle only when all the interior angles are 90 degrees

(1)  w=x  and  z=y  but  w≠z 
--> No information about the interior angles
--> It can be a parallelogram or rectangle -->  Insufficient

(2) Perimeter of the quadrilateral is  2(x+y) 
--> No information about the interior angles
--> It can be a parallelogram or rectangle -->  Insufficient

Combining (1) & (2),
No information about the interior angles.
--> It can be a parallelogram or rectangle -->  Insufficient

Option E

EgmatQuantExpert · Answer

Solution

Given  
In this question, we are given that
 • The diagram of a quadrilateral, with sides w, x, y, and z

To find  
We need to determine
 • Whether ABCD is a rectangle or not

Analysing Statement 1  
As per the information given in statement 1, w = x and z = y but w ≠ z

According to statement 1, ABCD is a parallelogram.
 • However, ABCD will be a rectangle only when the internal angles are 90.

As we don’t have any information about the internal angles, we cannot say whether ABCD is a rectangle or not.

Hence, statement 1 is not sufficient to answer the questions.

Analysing Statement 2  
As per the information given in statement 2, perimeter of the quadrilateral is 2(x + y).

This is possible when the opposite sides of ABCD are equal in length, and in such case ABCD can be either a parallelogram or a rectangle.
 • However, ABCD will be a rectangle only when the internal angles are 90.

As we don’t have any information about the internal angles, we cannot say whether ABCD is a rectangle or not.

Hence, statement 2 is not sufficient to answer the questions.

Combining Both Statements  
Even after combining both statements, we do not have any information about the internal angles of ABCD.

Thus, option E is the correct answer.

Correct Answer: Option E

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • w, x, y, and z are sides of quadrilateral ABCD.
• We need to find if ABCD is a rectangle.
 o For quadrilateral ABCD to be a rectangle, ABCD must satisfy following two conditions:
  Condition 1: Opposite sides must be equal.
 • i.e. w =x and z =y 
 Condition 2: all the interior angles must be right angles.   
With above analysis in mind, let’s analyse the question statements.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:   w=x   and  z=y  but  w≠z 
 • According to this statement, quadrilateral ABCD satisfy the condition 1 i.e. opposite sides are equal.
• However, we don’t know, if all the interior angles are 90 degree or not. 
So, we cannot conclude if ABCD is rectangle or not. It may be a parallelogram.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
 Statement 2:  Perimeter of the quadrilateral is 2(x+y)
 • According to this statement, perimeter of  ABCD = w+x+y+z = 2(x+y) 
 o This means two sides of quadrilateral ABCD are equals. It may be:
   Case 1 :  x = w  and  y = z  
 • In this case it may be a parallelogram. 
  Case 2 :  x = z  and  y =w . 
o However, we still cannot conclude anything about interior angles.   
Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.

Step 3: Analyse Statements by combining. 
 • From statement 1:  w = x  and  z = y  but  w≠z  
• From statement 2: 
 o  x = w  and  y = z  or
o  x = z  and  y =w  
• On combining both, we have:  w = x  and  z = y  but  w≠z  i.e opposite sides are equal and adjacent sides are not equal.
• However, we still don’t know if the interior angles are 90 degrees. 
Thus, the correct answer is  Option E.

Is quadrilateral ABCD with sides w, x, y, and z a rectangle?

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