### All GMAT Math Resources

## Example Questions

### Example Question #51 : Quadrilaterals

For rectangle PQRS, the ratio of PQ to QR is 2 to 3. What is the length of SQ?

(1) The length of PS is 9.

(2) The length of SR is 6.

**Possible Answers:**

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

**Correct answer:**

EACH statement ALONE is sufficient.

With a ratio of sides of 2:3, statement 1 tells us PS and QR are 9 units long. Knowing the ratio we can determine both PQ and SR are 6. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ: . Therefore, statement 1 alone is sufficient.

With a ratio of sides of 2:3, statement 2 tells us PQ and SR are 6 units long. Knowing the ratio we can determine both PS and QR are 9 units long. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ: . Therefore, statement 2 alone is sufficient.

Therefore, the correct answer is EACH statement ALONE is sufficient.

### Example Question #52 : Quadrilaterals

For parallelogram ABCD, what is the length of AC?

(1) BD is 16.

(2) AE is 9.

**Possible Answers:**

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

**Correct answer:**

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

By definition, a parallelogram is a quadrilateral with 2 sets of parallel sides and diagonals that bisect each other.

From statement 1 we cannot determine the length of AC, since ABCD is only defined as a parallelogram and not necessarily a rectangle (which is a special parallelogram with equal diagonals). Therefore, statement 1 alone is not sufficient.

From statement 2, we can determine that since the definition of a parallelogram states the diagonals bisect each other. Therefore, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

### Example Question #1 : Dsq: Calculating The Length Of The Diagonal Of A Quadrilateral

What is the length of PR?

(1) PQRS is a parallelogram

(2)

**Possible Answers:**

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

**Correct answer:**

Statements (1) and (2) TOGETHER are NOT sufficient.

With the information from statement 1, we don't know the measurements of any parts of PQRS. Therefore, Statement 1 alone is not sufficient.

With the information from statement 2, we know that , but since we have no confirmed definition of the shape of PQRS, we can't extrapolate the length of SQ to any other length. Therefore, statement 2 alone is not sufficient.

Even if we look at both statements 1 and 2 together, quadrilateral PQRS looks like a rectangle, but we are only told it is a parallelogram. The figures on the GMAT are not drawn to scale, therefore your eyes cannot be trusted. No difinitive length of PR can be determined. Therefore, statements 1 and 2 together are not sufficient.

Therefore, the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.

### Example Question #31 : Other Quadrilaterals

Consider rectangle .

I) Side is fathoms long.

II) Side is three fourths of .

What is the length segment , the diagonal of ?

**Possible Answers:**

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

**Correct answer:**

Both statements are needed to answer the question.

To find the diagonal of a rectangle, we need the length of two sides.

I) Gives us the length of one side.

II) Lets us find the length of the next side.

Use I) and II) with Pythagorean Theorem to find the diagonal.

Conversely, recognize that we are making a 3/4/5 Pythagorean Triple and see that the last side is 30 fathoms.

### Example Question #51 : Quadrilaterals

Find the diagonal of a square.

Statement 1: The perimeter of the square is known.

Statement 2: A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.

**Possible Answers:**

**Correct answer:**

Statement 1: The perimeter of the square is known.

A square has four equal sides. Write the perimeter formula for squares.

The side length of a square is a fourth of the perimeter.

Statement 2: A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.

If the primary square corners touch each of the secondary square edges, they must touch at the midpoint of each edge. Since Statement 2 mentions that the secondary square area is known, it is possible to solve for the edge length and the diagonal of the secondary square. Write the formula for the area of a square.

The diagonal of the secondary square can be solved by using the Pythagorean Theorem.

The side length of the secondary square also must equal the diagonal of the primary square.

Therefore:

### Example Question #51 : Quadrilaterals

What are the lengths of the diagonals in the parallelogram ?

1)

2)

**Possible Answers:**

Together, the two statements are sufficient.

Statement 1 alone is sufficient.

Neither of the statements, separate or together, is sufficient.

Either of the statements alone is sufficient.

Statement 2 alone is sufficient.

**Correct answer:**

Together, the two statements are sufficient.

Each of the statements, 1 and 2, provide something integral to calculating the length of a diagonal: an angle and the length of the sides connected to its vertex. The diagonal would be the third leg of the resulting triangle, and can be calculated using the law of cosines:

Since this is a parellelogram, knowing one angle allows us to know all the angles.

### Example Question #57 : Quadrilaterals

For the parellogram , the longest diagonal is . Is ?

1) The area of is

2) The perimeter of is

**Possible Answers:**

Statement 2 alone is sufficient.

Neither of the statements, separate or together, is sufficient.

Statement 1 alone is sufficient.

Together, the two statements are sufficient.

Either of the statements is sufficient.

**Correct answer:**

Statement 2 alone is sufficient.

Area alone is not enough information. Imagine, for instance, a parallelogram with a shorter side of and a longer side of . The diagonal would be well above

However, with the perimeter, the smaller and larger sides must add up to one half of it, .

The longer diagonal reaches its maximum with the larger internal angle widens towards degrees, and the parallelogram flattens into a line. Using the law of cosines, this translates to:

.

At its max, the diagonal could be no greater than

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