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.

Suggested GMAT Problem Solving Questions:

• The-set-s-of-numbers-has-the-following-properties
• If a = 1 and (a - b)/c = 1 which of the following is NOT a possible value of b?
• Walking at 6/7 th of his usual speed, a man is 25 min too late
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes Whereas a leak in the Bottom of the Cistern can Empty a Filled Tank in 40 minutes
• A milkman cheats his customers by adding water to the milk he sells
• if 80 lamps can be lighted, 5 hours per day for 10 days for $21.25, then the number of lamps,
• Few of the corporate contributions to the earthquake relief fund, aside from Pterocom
• The product of the first 10 prime numbers is closest to which of the following?
• There is a 120 liter mixture of alcohol and water. The ratio of alcohol to water is 7 : 5
• An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles
• A Furniture Store Sells Only Two Models of Desks, Model A and Model B. The Selling Price of Model A is $120
• A contractor combined x tons of a gravel mixture that contained 10 percent gravel G
• There are 100 Apples in a Bag of which 98% are Green and Rest are Red
• How many litres of a 90% solution of concentrated acid needs to be mixed with a 75% solution
• Jug Contains Water And Orange Juice In The Ratio 5:7. Another Jug Contains Water And Orange J
• The number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated
• A train travels from Albany to Syracuse, a distance of 120 miles, at an average rate of 50 miles per hour
• The hexagon ABCDEF is regular. That means all its sides are the same length and all its interior angles are the same size.
• y varies directly as x and when x = 6, y = 24. What is the value of y, when x = 5?
• If k is an Integer and 2 < k < 7, for How Many Different Values of k is There a Triangle With Sides of Lengths 2, 7, and k?
• How many factors does 36² have?
• A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively
• A Pentagon With 5 Sides Of Equal Length And 5 Interior Angles Of Equal measure is Inscribed in a Circle
• The Average Age of Chief Executive Officers (CEO’s) in a Large Sample of Companies is 57
• p, r, s, t, u An Arithmetic Sequence is a Sequence in Which each Term After the First Term is Equal to the Sum of the Preceding Term and a Constant
• If The Average (arithmetic mean) of The Four numbers 3, 15, 32, and (N + 1) is 18, then N =
• A Right Angled Triangle has its Sides in Arithmetic Progression and Being Integers
• Out of 7 Consonants and 4 Vowels, How Many Words of 3 Consonants and 2 Vowels Can be Formed?
• 4 Bells Toll Together at 9:00 A.M. They Toll After 7, 8, 11 and 12 seconds Respectively
• If 0 < a < b < c, which of the following statements must be true?
• 12 Marbles are Selected at Random from a Large Collection of White, Red, Green and Yellow Marbles
• What is the Remainder When 3³⁵ is Divided by 5?
• If The Sum of The 4th Term and The 12th Term of an Arithmetic Progress
• The symbol ∆ Denotes One of The Four Arithmetic Operations: Addition, Subtraction, Multiplication, or Division
• Find the Greatest Number That Will Divide 43,91 and 183 So as to Leave
• Six Bells Commence Tolling Together and Toll at Intervals of 2, 4, 6, 8 10 and 12 seconds Respectively
• Excluding Stoppages, The Speed of a Bus is 54 km/hr and Including Stoppage, It is 45 km/hr
• Find The Value Of x
• If y = x² - 6x + 9, what is the value of x?
• Is xy<1?
• S is a set of n consecutive positive integers. Is the mean of the set a positive integer?
• If y=2(x+1)y=2(x+1), What is the Value of y – x?
• From a group of M employees, N will be selected, at random, to sit in a line of N chairs
• The Selling Price of an Article is Equal to the Cost of the Article
• Does the equation y = (x – p)(x – q) intercept the x-axis at the point (2,0)?
• What is the value of x if x³ < x²?
• If Point O is The Centre of The Circle in The Figure Above, What is The Radius of The Circle?
• A merchant has 1000 kg of sugar part of which he sells at 8% profit and the rest at 18% profit
• If 30 is divided by half and 10 is added to the result, then what is the final result?
• Company C sells a line of 25 products with an average retail price of $1,200