GMAT Math : DSQ: Calculating whether rectangles are similar

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

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.

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:

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 #2 : Dsq: Calculating Whether Rectangles Are Similar

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:

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 insufficient 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.

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 #3 : 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 .

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.

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient 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 #4 : 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 insufficient to answer the question.

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

EITHER statement ALONE is sufficient 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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Example Question #5 : Dsq: Calculating Whether Rectangles Are Similar

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

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

Statement 1: $\frac{AC}{EG} = \frac{BD}{FH}$

Statement 2: $\frac{AB}{EF} = \frac{BC}{FG}$

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Statement 1 compares the lengths of the diagonals of the two rectangles. Since the diagonals of any rectangle are congruent, AC = BD , EG = FH , and, as a consequence, $\frac{AC}{EG} = \frac{BD}{FH}$ regardless of whether the rectangles are similar or not. Statement 1 is a superfluous statement and is therefore unhelpful.

Statement 2 asserts that sides of the two rectangles are in proportion. This is a necessary and sufficient condition for the rectangles to be similar.

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Example Question #6 : 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: $\left (\frac{p }{q} \right ) ^{2}= \frac{r}{s}$

Statement 2: $\frac{AC}{EG} = \frac{p}{q}$

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true.

Rectangles

Regardless of which $\textup{Rect } EFGH$ is chosen:

The ratio of the perimeters is $\frac{14}{28} = \frac{1}{2}$ ;

The ratio of the areas is $\frac{12}{48} = \frac{1}{4} =\left ( \frac{1 }{2} \right )^{2}$ ; and,

The ratio $\frac{AC}{EG} = \frac{5}{10} = \frac{1}{2}$ .

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 similarity statement given may be true or false).

The two statements together provide insufficient information.

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GMAT Math : DSQ: Calculating whether rectangles are similar

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

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