GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #4 : Calculating The Length Of The Diagonal Of A Quadrilateral

Calculate the length of the diagonal for a rectangle with a length of    and a width of  .

Possible Answers:

Correct answer:

Explanation:

The diagonal of a rectangle can be thought of as the hypotenuse of a right triangle whose base and height are the length and width of the rectangle, respectively.  This means we can use the Pythagorean Theorem to calculate the length of the diagonal for a rectangle:

Example Question #5 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has area 72.  The lengths of  and  are both whole numbers, and  is the longer diagonal. Which of the following could be the length of ?

Possible Answers:

None of the other choices gives a correct answer.

Correct answer:

Explanation:

The area of a rhombus is half the product of the lengths of its diagonals, which here are  and . This means

Since both diagonals have whole numbers as their lengths, and , we are looking for  to be a whole number that can be divided into 144 to yield a quotient less than . Equivalently, 

The quotient is 

Since we can multiply both sides by  to yield the inequality

,

we know that

so we can eliminate 12 and 16.

Also, since , 10 is not correct, as  would not be a whole number.

If , then . Both diagonals have lengths that are whole numbers, and  is the longer diagonal. 9 is the correct choice.

Example Question #6 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has perimeter 80; . What is the length of  ? 

Possible Answers:

Correct answer:

Explanation:

A rhombus has four sides of equal length. Since Rhombus  has a right angle , it follows that, the rhombus being a parallelogram, all four angles are right angles, and, by definition, Rhombus  is a square. The length of a diagonal of a square is  times the length of a side; since the rhombus has perimeter 80, each side measures one fourth of this, or 20, and diagonal  has length .

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

Rhombus  has area 56. 

Which of the following could be true about the values of  and ?

Possible Answers:

None of the other responses gives a correct answer.

Correct answer:

Explanation:

The area of a rhombus is half the product of the lengths of its diagonals, which here are  and . This means

Therefore, we need to test each of the choices to find the pair of diagonal lengths for which this holds.

 

:

Area: 

 

Area: 

 

Area: 

 

Area: 

 is the correct choice.

 

Example Question #51 : Rhombuses

Rhombus  has perimeter 64; . What is the length of  ?

Possible Answers:

Correct answer:

Explanation:

The sides of a rhombus are all congruent; since the perimeter of Rhombus  is 64, each side measures one fourth of this, or 16. 

The referenced rhombus, along with diagonal , is below:

 

Rhombus

Since consecutive angles of a rhombus, as with any other parallelogram, are supplementary,  and  have measure ;  bisects both into  angles, making equilangular and, as a consequence, equilateral. Therefore, .

Example Question #2 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has perimeter 48; . What is the length of  ? 

Possible Answers:

Correct answer:

Explanation:

The referenced rhombus, along with diagonals  and , is below.

Rhombus

The four sides of a rhombus have equal measure, so each side has measure one fourth of the perimeter of 48, which is 12.

Since consecutive angles of a rhombus, as with any other parallelogram, are suplementary,  and  have measure ; the diagonals bisect  and  into  and  angles, respectively, to form four 30-60-90 triangles.  is one of them; by the 30-60-90 Triangle Theorem, ,

and

.

Since the diagonals of a rhombus bisect each other, .

Example Question #1 : Kites

Given: Quadrilateral  such that   is a right angle, and diagonal  has length 24.

Give the length of diagonal .

Possible Answers:

None of the other responses is correct.

Correct answer:

Explanation:

The Quadrilateral  is shown below with its diagonals  and .

. We call the point of intersection :

Kite

The diagonals of a quadrilateral with two pairs of adjacent congruent sides - a kite - are perpendicular; also,  bisects the  and angles of the kite. Consequently,  is a 30-60-90 triangle and  is a 45-45-90 triangle. Also, the diagonal that connects the common vertices of the pairs of adjacent sides bisects the other diagonal, making  the midpoint of . Therefore, 

.

By the 30-60-90 Theorem, since  and  are the short and long legs of 

By the 45-45-90 Theorem, since  and  are the legs of a 45-45-90 Theorem, 

.

The diagonal  has length 

.

Example Question #1 : Calculating Whether Quadrilaterals Are Similar

Which of the following rectangles is similar to one with a length of    and a width of   ?

Possible Answers:

Correct answer:

Explanation:

In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can check which dimensions are those of a rectangle similar to the given one by first calculating the ratio of the length to the width for the given rectangle, and then doing the same for each of the answer choices until we find which has an equal ratio between its dimensions:

So in order for a rectangle to be similar to the given rectangle, this must be the ratio of its length to its width. Now we check the answer choices, in no particular order, for one with this ratio:

We can see that only the rectangle with a length of    and a width of    has the same ratio as the given rectangle, so this is the similar one.

Example Question #452 : Problem Solving Questions

Which of the following dimensions would a rectangle need to have in order to be similar to one with a length of    and a width of   ?

Possible Answers:

Correct answer:

Explanation:

In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can calculate the ratio of length to width for the given rectangle, and then check the answer choices for the rectangle whose dimensions have the same ratio:

Now we check the answer choices, in no particular order, and the dimensions with the same ratio are those of the rectangle that is similar:

We can see that a rectangle with a length of    and a width of    has the same ratio of dimensions as the given rectangle, so this is the one that is similar.

Example Question #1 : Calculating Whether Quadrilaterals Are Similar

Trapezoid

Refer to the above Trapezoid . There exists Trapezoid  such that

Trapezoid  Trapezoid , and the length of the midsegment of Trapezoid  is 91. 

Give the length of .

Possible Answers:

Correct answer:

Explanation:

The length of the midsegment of a trapezoid - the segment that has as its endpoints the midpoints of its legs - is half the sum of the lengths of its legs. Therefore,  Trapezoid  has as the length of its midsegment

.

Sidelengths of similar figures are in proportion. If the similarity ratio is , then the bases of Trapezoid  have length  and , so their midsegment will have length

,

meaning that the ratio of the lengths of the midsegments will be the same as the similarity ratio. Since the length of the midsegment of Trapezoid  is 91, this similarity ratio is

 .

The ratio of the length of  to that of corresponding side  is therefore , so

 

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