GMAT Math : Geometry

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Example Questions

Example Question #121 : Geometry

What is the perimeter of rectangle ABCD ?

(1) The area of ABCD is 81

(2) Side AB=9

Possible Answers:

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

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

Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

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

EACH statement ALONE is sufficient.

Correct answer:

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

Explanation:

To know the perimeter of a rectangle, we need to know length and width.

Statement (1) provides Area, which does not provide the necessary information. NOT SUFFICIENT

Statement(2) provides one side, but we still need the other to determine perimeter. NOT SUFFICIENT.

Both together are sufficent however since if we know one side and the area, we can find the remaining side using division, and then use the two side lengths to find the perimeter.

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Example Question #2 : Dsq: Calculating The Perimeter Of A Rectangle

You are given a square and a rectangle. Which has the greater perimeter?

Statement 1: The length of the rectangle is twice the sidelength of the square,

Statement 2: The width of the rectangle is half the sidelength of the square.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Let be the sidelength of the square. Then its perimeter is .

From Statement 1 alone, the length of the rectangle is , and, if is the width, the perimeter of the rectangle is 2s + 2s + w + w = 4s + 2w > 4s . Therefore, we can prove that the rectangle has the greater perimeter.

From Statement 2 alone, the width of the rectangle is $\frac{1}{ 2}s$ , and its perimeter is at least - but unless we know the length, we do not know whether the total perimeter is greater than, equal to, or less than .

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Example Question #3 : Dsq: Calculating The Perimeter Of A Rectangle

The table in a hall has a length of feet. What is its perimeter?

I) The tabletop is exactly two and a half feet above the floor.

II) The area of the table is four times three more than the width.

Possible Answers:

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

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

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:

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

Explanation:

To find perimeter we need length and width. We are given the length.

I) Is irrelevant.

II) We are given a way of relating area and width. Since we know that area is length times width, we can use II to set up an equation where we substitute in the known length along with the given statement to solve for our width

$A=l\cdot w$

$A=4\cdot (w+3)= 14w$

Solve the second one for w and you're good to go!

A=4w+12=14w

12=10w

$\frac{12}{10}=\frac{6}{5}=w$ .

Therefore the perimeter would be:

$P=2l+2w \rightarrow P=2(14)+2\left(\frac{6}{5}\right)$

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Example Question #4 : Dsq: Calculating The Perimeter Of A Rectangle

Farmer Jeff Jenkins is making a new rectangular field for his goats and needs to know how much fencing he needs to buy.

I) The goats will need at least 640 square yards to roam.

II) One edge of the field will be yards long and will be made up of a river.

Possible Answers:

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

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

Either statement is sufficient to answer the question.

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

Both statements are needed to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

To find perimeter, we need the lengths of both sides.

I) Gives us the area, which gives us one equation and two unknowns. Alone this statement is not sufficient.

II) Gives us one side length of the field.

We can use this with I) to find the other side length.

Once we have both sides, we can find perimeter easily.

$A=bh \rightarrow 640=40h$

h=16

Thus the perimeter is,

$P=2b+2h \rightarrow P=2(40)+2(16)=112$ .

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Example Question #2 : Dsq: Calculating The Perimeter Of A Rectangle

What is the perimeter of the rectangle?

The area is .
The width measures .

Possible Answers:

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Each statement alone is sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Explanation:

Statement 1: Although the area is given, we need more information in order to answer the question.

$l\times w=12$ but this can means our dimenions are $6 \times 2$ or $4 \times 3$

Statement 2: We're told the width measures which means our dimensions are $6\times 2$ .

Using BOTH statements, we can find the perimeter: 2(l+w)=2(2+6)=16

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Example Question #6 : Dsq: Calculating The Perimeter Of A Rectangle

True or false: A+ B > 50

Statement 1: A rectangle with length and width has area greater than 100.

Statement 2: A rectangle with length and width has perimeter greater than 100.

Possible Answers:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Statement 1 alone provides insufficient information. For example:

Case 1: A = 10, B = 12

The area of a rectangle with these dimensions is their product, which is 120; this exceeds 100.

A + B = 10+12 = 22 < 50

Case 2: A =40, B = 12

The area of a rectangle with these dimensions is their product, which is 480; this exceeds 100.

A + B = 40+12 = 52 > 50

Assume Statement 2 alone. The perimeter of a rectangle is twice the sum of their dimensions, so

2(A+B)> 100

$\frac{2(A+B)}{2}> \frac{100}{2}$

A + B > 50 , answering the question in the affirmative.

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Example Question #21 : Rectangles

Rectangles

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows a rectangle divided into two smaller rectangles.

True or false: $\textup{Rect } ABCF \sim \textup{Rect } CDEF$ .

Statement 1: $CF = \sqrt{AF \cdot FE}$

Statement 2: $AF \cdot FE = AB \cdot DE$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The statements are actually equivalent. Since CF = AB= DE , Statement 2 can be rewritten as $AF \cdot FE = CF \cdot CF$ , or $\left (CF \right )^{2} =AF \cdot FE$ - and, since all quantities are positive, $CF = \sqrt{AF \cdot FE}$ . The question is therefore whether either statement alone answers the question or both together do not.

Each statement is equivalent to

$AF \cdot FE = CF \cdot CF$ .

Divide both sides by $FE \cdot CF$ to yield a proportion statement:

$\frac{AF \cdot FE}{FE \cdot CF} = \frac{CF \cdot CF}{FE \cdot CF}$

$\frac{AF }{ CF} = \frac{CF }{FE }$

The sides of the rectangles are in proportion; subsequently, the rectangles are similar.

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Example Question #22 : Rectangles

You are given two rectangles, $\textup{Rect } ABCD$ and $\textup{Rect } EFGH$

True or false: $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

Statement 1: $AB \cdot FG= BC \cdot EF$

Statement 2: $AB \cdot GH= CD \cdot EF$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. Divide both sides by $EF \cdot FG$ ,

$AB \cdot FG= BC \cdot EF$

$\frac{AB \cdot FG}{EF \cdot FG}= \frac{BC \cdot EF}{EF \cdot FG}$

$\frac{AB}{EF} = \frac{BC}{FG}$ .

This proportion statement asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.

Now examine Statement 2.

By the congruence of opposite sides of a rectangle,

AB = CD , GH = EF ,

and, regardless of whether the rectangles are similar or not,

$AB \cdot GH= CD \cdot EF$ .

Therefore, Statement 2 provides superfluous and unhelpful information.

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Example Question #81 : Quadrilaterals

You are given two rectangles, $\textup{Rect } ABCD$ and $\textup{Rect } EFGH$ .

Let the perimeter of $\textup{Rect } ABCD$ be , and let the perimeter of $\textup{Rect } EFGH$ be .

True or false: $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

Statement 1: $\frac{AB}{EF} = \frac{p}{q}$

Statement 2: $\frac{AD}{EH} = \frac{p}{q}$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The perimeter of $\textup{Rect } ABCD$ is the sum of its sidelengths, and opposite sides are congruent, so

p = AB + BC + CD + AD

p = AB + BC + AB+ BC

$p = 2 \cdot AB + 2\cdot BC$

Similarly,

$q = 2 \cdot EF + 2 \cdot FG$

Therefore,

$\frac{p }{q}= \frac{2 \cdot AB + 2\cdot BC}{2 \cdot EF + 2 \cdot FG}$ ,

and, reducing,

$\frac{p }{q}= \frac{ AB + BC}{ EF + FG}$

Assume Statement 1 alone. Then

$\frac{p}{q} = \frac{AB}{EF}$ , or $\frac{p}{q} = \frac{-AB}{-EF}$ , and by a property of proportions,

$\frac{p }{q}= \frac{\left ( AB + BC \right )+(-AB)}{ \left (EF + FG \right )+(-EF)} = \frac{BC}{EF}$

Therefore,

$\frac{AB}{EF}= \frac{BC}{EF}$ ,

thereby proving the sides of the rectangles to be in proportion. As a consequence, $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

By a similar argument, Statement 2 also proves $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

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Example Question #1 : Dsq: Calculating Whether Rectangles Are Similar

You are given two rectangles, $\textup{Rect } ABCD$ and $\textup{Rect } EFGH$ .

Let the perimeter of $\textup{Rect } ABCD$ be , and let the perimeter of $\textup{Rect } EFGH$ be .

Let the area of $\textup{Rect } ABCD$ be and the area $\textup{Rect } EFGH$ be .

True or false: $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

Statement 1: $\frac{p^{2}}{q^{2}} = \frac{r}{s}$

Statement 2: $\frac{AB}{EF} = \frac{p}{q}$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone.

Examine these three rectangles. The one of the left is $\textup{Rect } ABCD$ ; the other two have the same dimensions, and both are called $\textup{Rect } EFGH$ , except that the names of the vertices are differently arranged:

Rectangles

Regardless of which $\textup{Rect } EFGH$ is chosen, the ratio of the perimeters is $\frac{14}{28} = \frac{1}{2}$ and the ratio of the areas is $\frac{12}{48} = \frac{1}{4} =\left ( \frac{1 }{2} \right )^{2}$ . The conditions of the problem are met for both pairings, but in one case, $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ and in the other, $\textup{Rect } ABCD \nsim \textup{Rect } EFGH$ (that is, the rectangles are similar, but the given similarity statement may or may not be true).

Assume Statement 2 alone.

The perimeter of $\textup{Rect } ABCD$ is the sum of its sidelengths, and opposite sides are congruent, so

p = AB + BC + CD + AD

p = AB + BC + AB+ BC

$p = 2 \cdot AB + 2\cdot BC$

Similarly,

$q = 2 \cdot EF + 2 \cdot FG$

Therefore,

$\frac{p }{q}= \frac{2 \cdot AB + 2\cdot BC}{2 \cdot EF + 2 \cdot FG}$ ,

and, reducing,

$\frac{p }{q}= \frac{ AB + BC}{ EF + FG}$

From Statement 2,

$\frac{p}{q} = \frac{AB}{EF}$ , or $\frac{p}{q} = \frac{-AB}{-EF}$ , and by a property of proportions,

$\frac{p }{q}= \frac{\left ( AB + BC \right )+(-AB)}{ \left (EF + FG \right )+(-EF)} = \frac{BC}{EF}$

Therefore,

$\frac{AB}{EF}= \frac{BC}{EF}$ ,

thereby proving the sides of the rectangles to be in proportion. As a consequence, $\textup{Rect } ABCD \sim \textup{Rect } EFGH$ .

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #121 : Geometry

Example Question #2 : Dsq: Calculating The Perimeter Of A Rectangle

Example Question #3 : Dsq: Calculating The Perimeter Of A Rectangle

Example Question #4 : Dsq: Calculating The Perimeter Of A Rectangle

Example Question #2 : Dsq: Calculating The Perimeter Of A Rectangle

Example Question #6 : Dsq: Calculating The Perimeter Of A Rectangle

Example Question #21 : Rectangles

Example Question #22 : Rectangles

Example Question #81 : Quadrilaterals

Example Question #1 : Dsq: Calculating Whether Rectangles Are Similar

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