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555-605 (Medium)|   Geometry|      
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Is quadrilateral ABCD with sides w, x, y, and z a rectangle? 30 Mar 2020, 02:38
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E
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25% (medium)

Question Stats:

67% (00:54) correct 33% (01:03) wrong based on 63 sessions

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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 \(2(x+y)\)


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E
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Is quadrilateral ABCD with sides w, x, y, and z a rectangle? 30 Mar 2020, 02:55
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Bunuel

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 \(2(x+y)\)
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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
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Is quadrilateral ABCD with sides w, x, y, and z a rectangle? 30 Mar 2020, 03:56
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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
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Is quadrilateral ABCD with sides w, x, y, and z a rectangle? 30 Mar 2020, 05:33
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Bunuel

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 \(2(x+y)\)
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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.
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