Is Quadrilateral ABCD a Rectangle? GMAT Data Sufficiency

Question: Is quadrilateral ABCD a rectangle?

     (1) Line segments AC and BD bisect one another. 

     (2) Angle ABC is a right angle.

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.

Correct Answer: C

Solution and Explanation:
Approach Solution 1:
It is asked in the question whether a quadrilateral ABCD is a rectangle or not.

It should be noted that every quadrilateral might not be a rectangle but every rectangle is a quadrilateral.

The given question statement can be rephrased

Does a quadrilateral ABCD has

i) all the opposite sides equal

ii) opposite sides parallel

iii) Bisecting diagonals

iv) adjacent sides making 90 degrees with each other

v) adjacent angles that sum to 180 degrees.

If all the above conditions are satisfied then it is sufficient to say that the given quadrilateral is a rectangle.

Firstly we’ll check statement 1.

1. Line segment AC and BD bisect each other

If the line segments AC and BD bisect each other then it is sufficient to say that the quadrilateral ABCD is a parallelogram. But it is not sufficient to say that ABCD is a rectangle.

Now check statement 2.

2. Angle ABC is a right angle.

One of the angles in the quadrilateral is 90 degrees. But only one right angle does not necessarily make ABCD a rectangle. This statement alone is not sufficient to say that ABCD is a rectangle.

Combining both statements we get:
As we know a parallelogram with one right angle has all right angles. It is because opposite angles are equal and complementary angles are 180 degrees.

Hence, both statements together are sufficient. 

Approach Solution 2:
The problem statement asks whether a quadrilateral ABCD is a rectangle or not.

From the given statements,

Statement one is sufficient to say that ABCD is a parallelogram.

Statement 2 says that one of the angles (B here) is 90 degrees.

One of the useful properties of a parallelogram: the adjacent angles of a parallelogram are supplementary. (supplementary angles are angles whose sum adds up to 180). So if angle B is 90 degrees angles A And C must be 90 degrees.

This implies that all the angles are 90 degrees in the parallelogram.

It is also known that a parallelogram has all opposite sides parallel and equal.

Also, all the angles in the parallelogram are 90 degrees.

From statements 1 and 2, it is sufficient to say that the shape ABCD is either a square or a rectangle. Either way, both are rectangles.

Hence, both statements together are sufficient.

Approach Solution 3:
The problem statement asks whether a quadrilateral ABCD is a rectangle or not.

Statement 1: Line segments AC and BD bisect one another. 
Therefore it implies that the diagonals bisect each other.
Hence, it is a parallelogram. For ABCD to be a rectangle, both diagonals need to be equal.
However, it is nowhere that diagonals are equal or not. Hence, statement one alone is not sufficient.

Statement 2: Angle ABC is a right angle.
This implies angle ABC= 90 degree
Hence, statement two alone is not sufficient.

Combining both statement 1 & statement  2, we get:
ABCD is a parallelogram with one angle of 90 degrees.
This implies that it has to be a rectangle and both diagonals become automatically equivalent.
Hence, both statements together are sufficient.

“Is quadrilateral ABCD a rectangle”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Prep Plus”. These quantitative questions help the candidates to improve their mathematical knowledge and understanding. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Suggested GMAT Data Sufficiency Questions

• If abc 0, is a < b < c? (1) 1/c < 1/b < 1/a (2) c > a GMAT Data Sufficiency
• A Recent Lunch Meeting at a Certain Club was Attended by Members GMAT Data Sufficiency
• w, x, y, and z are integers. If w > x > y > z > 0, is y a common GMAT Data Sufficiency
• What is the sum of 3 consecutive integers? GMAT Data Sufficiency
• Two cars A and B start from diametrically opposite points of GMAT Data Sufficiency
• What Is The Value Of X + Y In The Figure Above? GMAT Data Sufficiency
• The Length, Width, and Height of a Rectangular Box, in Centimeters are L, W, and H GMAT Data Sufficiency
• Is the perimeter of a rectangle greater than 8 inches GMAT Data Sufficiency
• Is the Positive Integer n Divisible by 8? (1) n is Divisible by the Product of Three GMAT Data Sufficiency
• Two Cars, A and B, Each Traveled a Distance of 200 Miles GMAT Data Sufficiency
• What is the SD of a Given Set of Numbers Whose Average is 5? GMAT Data Sufficiency
• Is xy > 5 GMAT Data Sufficiency
• A Hotel has Two Types of Rooms: Basic and Deluxe. On a Certain Night GMAT Data Sufficiency
• The Sequence s1, s2, s3, ..., sn, ... is Such ThatSn= 1/n − 1/n + 1 GMAT Data Sufficiency
• At The Bakery, Lew Spent A Total Of $6.00 For One Kind Of GMAT Data Sufficiency
• If p, q, x, y And z Are Different Positive Integers, Which Of The Five GMAT Data Sufficiency
• Members Of A Wrestling Team Were Weighed Twice. Was The Standard GMAT Data Sufficiency
• What Is The Number Of Female Employees In Company X ? GMAT Data Sufficiency
• If k is a Positive Integer and n = k(k + 7), is n Divisible by 6? GMAT Data Sufficiency
• In the xy Plane, What is the Slope of Line l GMAT Data Sufficiency
• The Side Of An Equilateral Triangle Has The Same Length As The Diagonal GMAT Data Sufficiency
• Each of the Students in a Certain Class Received a Single Grade of P GMAT Data Sufficiency
• Does The Graphical Representation Of The Quadratic Function GMAT Data Sufficiency
• What is the Value of m? (1) |m| = −36/m (2) 2m + 2|m| = 0 GMAT Data Sufficiency
• Do The Diagonals Of A Quadrilateral ABCD Bisect Each Other GMAT Data Sufficiency
• If X = 0.rstu, Where r, s, t, And u Each Represent A Nonzero Digit GMAT Data Sufficiency
• Is The Product Of a And b Equal To 1? GMAT Data Sufficiency
• Material A Costs $3 Per Kilogram, And Material B Costs $5 Per Kilogram GMAT Data Sufficiency
• Is the Product of a Positive and a Negative Integer Less Than -10? GMAT Data Sufficiency
• If k is not Equal to 0, 1, or -1 is 1/k>0? GMAT Data Sufficiency
• If x Is A Positive Integer, What Is The Value Of (x+24)^(½) − x^(½) GMAT Data Sufficiency
• Points x And y Lie On A Line Segment. How Many Units Long Is The Line GMAT Data Sufficiency
• Which Sport Utility Vehicle has a Higher List Price, the Touristo or GMAT Data Sufficiency
• What is The Average (arithmetic mean) Annual Salary of the 6 Employees GMAT Data Sufficiency
• If S is a Set of Ten Consecutive Integers, is the Integer 5 GMAT Data Sufficiency
• What is the Perimeter of a Square Chessboard, If a Chessboard is made GMAT Data Sufficiency
• Is the sum of four different positive prime integers an odd integer? GMAT Data Sufficiency
• Is Positive Integer x Divisible By 24? GMAT Data Sufficiency
• 10 Students took a Chemistry Exam that was Graded on a Dcale of 0 to 100 GMAT Data Sufficiency
• If m and n are positive integers, is the remainder of GMAT Data Sufficiency
• In quadrilateral ABCD, is AC longer than BD GMAT Data Sufficiency
• N is a positive integer. Is n the square of an integer? GMAT Data Sufficiency
• If z Is An Integer Smaller Than Integer n But Greater Than Zero GMAT Data Sufficiency
• If y Is An Odd Number And x^3 ∗ y^4 = 648, y=? 1) y Is Positive 2) x Is Gmat Data Sufficiency
• A Pyramid-Shaped Box To Protect A Plant Is Constructed With 4 Lateral Faces GMAT Data Sufficiency
• Is abc/d an Integer if a, b, c, and d are Positive integers? GMAT Data Sufficiency
• If n is a positive integer, is (1/10)^n < 0.01? GMAT Data Sufficiency
• If m and n are nonzero integers, is m^n an integer? GMAT Data Sufficiency
• In A Certain Office, 50 Percent Of The Employees Are College GMAT Data Sufficiency
• As Shown Above, A Rectangle Is Divided Into Smaller Rectangles Of The GMAT Data Sufficiency