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

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 \(2(x+y)\)

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.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

• 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.

• 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.

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.