Is Quadrilateral ABCD A Rhombus? GMAT Data Sufficiency

Question: Is quadrilateral ABCD a rhombus?

(1) Line segments AC and BD are perpendicular bisectors of each other.
(2) AB = BC = CD = AD

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.

“Is quadrilateral ABCD a rhombus?”- is a topic that belongs to the GMAT Quantitative reasoning section of GMAT. The GMAT Quant section encloses 31 multiple-choice questions that need to be completed within 62 minutes.GMAT Data Sufficiency questions include a problem statement and two factual statements. This specific GMAT data sufficiency question estimates the talents of the candidates in cracking mathematical problems. The mathematical problems include mathematical reasoning, interpreting graphic data and solving quantitative problems. The difficult part of these question types is the clever wording that candidates tend to overlook. GMAT data sufficiency comes up with 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

The problem statement asks the candidate to determine whether quadrilateral ABCD is a rhombus.

1. The statement suggests that the line segments AC and BD are perpendicular bisectors of each other.
   Since the characteristics of a rhombus imply that the diagonals of a rhombus are perpendicular bisectors to each another.
   Therefore, this statement alone is sufficient as it offers valid information to prove that the quadrilateral ABCD is a rhombus.
2. The statement indicates that AB = BC = CD = AD. This means the sides of the rhombus are equal to each other.
   According to the definition of the rhombus, the quadrilateral possessing four equal sides is a rhombus.
   Therefore, this statement is sufficient to prove that the quadrilateral is a rhombus.

Hence, we can say, each of the statements alone is sufficient to demonstrate quadrilateral ABCD is a rhombus.

Correct Answer: D

Approach Solution 2:

The problem statement asks to resolve quadrilateral ABCD as a rhombus.

1. The statement recommends that the line segments AC and BD are perpendicular bisectors to one another.
   Let’s assume the point of intersection of the bisectors is X.
   Then as per the rule, BX = DX and AX = CX
   By applying the Pythagoras theorem,
   Let’s imagine the hypotenuse of the sides are BX and CX
   Since the value of AX is equal to the value of CX, the hypotenuse AX and BX is as long as the prior.
   Thus all the hypotenuses that formed are equal to each other, consequently forming four equal sides.
   As we know that a rhombus can be defined as a quadrilateral with each of its sides equal to one another. Therefore, it is proved that it is a rhombus since all sides are equal.
   Hence, the statement alone is sufficient to prove quadrilateral ABCD is a rhombus.
2. The statement implies that AB = BC = CD = AD. This represents that all the sides of the rhombus are equal to each other.
   The definition of rhombus suggests that the rhombus is a type of quadrilateral with each of its sides equal to one other.

Therefore, the statement alone is sufficient to prove quadrilateral ABCD is a rhombus.

Correct Answer: D

Approach Solution 3:

The problem statement asks to find out whether quadrilateral ABCD is a rhombus.

1. The statement states that the line segments AC and BD are perpendicular bisectors to one another. This implies that AC cuts the line BD into two equal halves at a right angle i.e. 90°. Similarly, line BD cuts the line AC into two equal halves at a right angle i.e. 90°.
   As per the rule, the perpendicular bisectors mean all sides and angles are equal to one another. The definition of rhombus suggests that the rhombus is a sort of quadrilateral with all its sides equal to one other.
   Therefore, the statement alone is sufficient to prove quadrilateral ABCD is a rhombus.
2. The statement hints that AB = BC = CD = AD. This symbolises that each of the sides of the rhombus is equal to the other.
   As per the definition of a rhombus, a quadrilateral with all its sides equal is a rhombus.
   Hence the statement alone is sufficient to prove quadrilateral ABCD is a rhombus.

Correct Answer: D

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