Is quadrilateral a square? GMAT data sufficiency

Question: Is quadrilateral a square?

(1) Three angles of a quadrilateral are 90 degrees each.
(2) 3 sides of a quadrilateral are equal in length.

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Answer: C

Solution and Explanation:

Approach Solution 1:
#1: Even though all four sides are equal, a square may still not exist. For instance, it is a rhombus if none of the angles are 90°. It's not enough.
#2: If there are three 90° angles, then the fourth angle must also be 90°. So does a quadrilateral that has all four angles at 90* indicate that it is a square? No. It may resemble a rectangle. It's not enough.
Combining #1 and #2: Each of the quadrilateral's four angles must be 90 degrees. A pair of opposite sides must be equal if all four angles are 90 degrees, therefore if three sides are equal, the fourth one must also be equal to these three. The quadrilateral must therefore be a square.
Correct option: C

Approach Solution 2:
(1) Three angles of a quadrilateral are 90 degrees each:
If a quadrilateral has three angles of 90 degrees each, we can conclude that it is a rectangle, but not necessarily a square. A rectangle has two pairs of opposite sides that are parallel and equal in length, but its four angles are not necessarily equal.
(2) Three sides of a quadrilateral are equal in length:
If a quadrilateral has three sides of equal length, we cannot conclude that it is a square. It could be a rhombus, which also has four sides of equal length, but its angles are not necessarily 90 degrees.
If both cases are true, then the quadrilateral must be a square.
A square is a type of rectangle with four equal sides and four right angles. If three angles of a quadrilateral are 90 degrees each, then it is a rectangle. If three sides of a quadrilateral are equal in length, then it is a rhombus. Therefore, a quadrilateral that is both a rectangle and a rhombus must be a square.
In a square, all four angles are 90 degrees, and all four sides are equal in length. Therefore, if both cases are true, we can conclude that the quadrilateral is a square.
Correct option: C

Approach Solution 3:
If we know that a quadrilateral has three angles of 90 degrees each, we can conclude that it is a rectangle. However, a rectangle is not necessarily a square, as it may have two pairs of parallel sides of different lengths. Therefore, statement (1) alone is not sufficient to determine whether the quadrilateral is a square.
If we know that three sides of a quadrilateral are equal in length, we cannot conclude that it is a square. A square is a special type of rectangle in which all four sides are equal in length. Therefore, statement (2) alone is not sufficient to determine whether the quadrilateral is a square.
Combining both statements, we can conclude that the quadrilateral is a square since it has four equal sides and four right angles.
Therefore, the answer is (C) Both statements taken together are sufficient, but neither statement alone is sufficient.
Correct option: C

“Is quadrilateral a square? GMAT data sufficiency" - is a topic of the GMAT data sufficiency section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT data sufficiency questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and two statements. By using mathematics to answer the question, the candidate must select the appropriate response among five choices which states which statement is sufficient to answer the problem. The data sufficiency section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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