Question

Is parallelogram PQRS a rhombus?
I. PQ=QR=RS=SP
II. The line segments SQ and RP are perpendicular bisectors of each other.

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II together are sufficient to answer the question but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

Memorizing the specific properties that define different quadrilaterals is key to solving geometry-based Data Sufficiency questions. For a parallelogram to be a rhombus, you need one of two conditions: either all four sides are equal, or the diagonals are perpendicular.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a Data Sufficiency question based on the properties of quadrilaterals. A rhombus is a specific type of parallelogram. We need to determine if the given statements provide enough information to definitively say "yes" or "no" to the question.
Definition of a Rhombus: A parallelogram with all four sides of equal length.
Properties of a Rhombus:

All properties of a parallelogram apply.
All four sides are congruent.
The diagonals are perpendicular bisectors of each other.
Any parallelogram that satisfies property 2 or 3 is a rhombus.
Step 2: Detailed Explanation:
Analyze Statement I: "PQ=QR=RS=SP"
This statement says that all four sides of the parallelogram PQRS are equal. By definition, a parallelogram with four equal sides is a rhombus. This statement alone is sufficient to answer the question with a definitive "yes".
Analyze Statement II: "The line segments SQ and RP are perpendicular bisectors of each other."
This statement describes the diagonals of the parallelogram. One of the key properties that distinguishes a rhombus from other parallelograms is that its diagonals are perpendicular. If the diagonals of a parallelogram are perpendicular bisectors of each other, the parallelogram must be a rhombus. This statement alone is also sufficient to answer the question with a definitive "yes".
Step 3: Final Answer:
Since each statement alone is sufficient to determine that the parallelogram is a rhombus, the correct answer is (D).