Question

Is parallelogram PQRS a rhombus?
(1) \( PQ = QR = RS = SP \)
(2) The line segments SQ and RP are perpendicular bisectors of each other.

• A. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
• B. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
• C. Both statements 1 and 2 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 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

When identifying rhombuses, remember that both equal sides and perpendicular diagonals are key characteristics.

Answer 1

The Correct Option is C

Step 1: Analyze Statement 1.
Statement 1 tells us that \( PQ = QR = RS = SP \), which means that all four sides of the parallelogram are equal. This is a characteristic of a rhombus, but we need to confirm whether the angles are 90 degrees to establish that it is a rhombus. Therefore, statement 1 alone is insufficient.
Step 2: Analyze Statement 2.
Statement 2 tells us that the diagonals \( SQ \) and \( RP \) are perpendicular bisectors of each other. This property is characteristic of a rhombus, as rhombuses have perpendicular bisecting diagonals. However, we still need to know that the sides are equal. Hence, statement 2 alone is also insufficient.
Step 3: Combine Both Statements.
Combining both statements, we know that the sides of the parallelogram are equal and the diagonals bisect each other perpendicularly. These two properties together confirm that the figure is a rhombus.
Step 4: Conclusion.
Both statements 1 and 2 together are sufficient to conclude that PQRS is a rhombus. So, the correct answer is (C).