GMAT Math : Calculating whether rectangles are similar

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Calculating Whether Rectangles Are Similar

Rectangles

Note: Figure NOT drawn to scale.

Refer to the above figure. $\textup{Rect }ABCD \sim \textup{Rect }WXYZ$ .

$\frac{AD}{WZ} = \frac{7}{5}$ .

Give the ratio of the area of the shaded region to the area of $\textup{Rect }WXYZ$ .

Possible Answers:

24:25

5:7

7:5

1:1

25:24

Correct answer:

24:25

Explanation:

The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of $\textup{Rect }ABCD$ to that of $\textup{Rect }WXYZ$ is

$\left ( \frac{7}{5} \right )^{2} = \frac{49}{25}$ .

So if the area of $\textup{Rect }WXYZ$ is , the area of $\textup{Rect }ABCD$ is $\frac{49}{25} N$

The area of the shaded region is the difference between the areas of the rectangles, making this area

$\frac{49}{25} N - N = \frac{24}{25} N$ .

The desired ratio is 24 to 25.

Report an Error

Example Question #2 : Calculating Whether Rectangles Are Similar

$\textup{Rect }ABCD \sim \textup{Rect }WXYZ$ ;

$\frac{AB}{WX} = 3$ .

Which of the following must be equal to $\frac{1}{3}$ ?

Possible Answers:

The perimeter of $\textup{Rect }ABCD$ divided by the perimeter of $\textup{Rect }WXYZ$ .

$\frac{WY}{BD}$

None of the other choices gives a correct answer.

$\frac{WZ}{AB}$

The area of $\textup{Rect }WXYZ$ divided by the area of $\textup{Rect }ABCD$

Correct answer:

$\frac{WY}{BD}$

Explanation:

$\frac{AB}{WX} = 3$ , so $\frac{WX}{AB} = \frac{1}{3}$ ; this is the similarity ratio of $\textup{Rect }WXYZ$ to $\textup{Rect }ABCD$ .

$\overline{WZ}$ and $\overline{AB}$ are not corresponding sides, nor are they congruent to corresponding sides, so it may or may not be true that $\frac{WZ}{AB} = \frac{1}{3}$ .

The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not $\frac{1}{3}$ .

The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is $\left (\frac{1}{3} \right )^{2} = \frac{1}{9}$ .

However, by similarity, and the fact that opposite sides $\overline{BD }$ and $\overline{AC}$ are congruent,

$\frac{WY}{BD} = \frac{WY}{AC} = \frac{WX}{AB} = \frac{1}{3}$ .

The correct choice is $\frac{WY}{BD}$ .

Report an Error

Example Question #3 : Calculating Whether Rectangles Are Similar

Rectangles

In the above diagram,

$\textup{Rect }ABCF \sim \textup{Rect }CDEF$ .

CF = 24 and EF = 12 . Give the area of $\textup{Rect }ABDE$ .

Possible Answers:

720

288

576

1,152

1,440

Correct answer:

1,440

Explanation:

$\textup{Rect }ABCF \sim \textup{Rect }CDEF$ , so

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

$\frac{AF}{24} = \frac{24}{12}$

$AF = \frac{24}{12} \cdot 24 = 48$

The area of the rectangle is

$AE \cdot DE$

$= \left (AF + EF \right ) \cdot CF$

$= \left (48+12 \right ) \cdot 24$

$= 60 \cdot 24$

= 1,440

Report an Error

Example Question #4 : Calculating Whether Rectangles Are Similar

$\textup{Rect }ABCD \sim \textup{Rect }WXYZ$ , and $\frac{AB}{WX} = 2$ .

All the following quantities MUST be equal to 2 except for __________.

Possible Answers:

$\frac{AC}{WY}$

$\frac{BC}{WZ}$

The perimeter of $\textup{Rect }ABCD$ divided by the perimeter of $\textup{Rect }WXYZ$ .

All of the quantities in the other choices must be equal to .

$\frac{CD}{YX}$

Correct answer:

$\frac{CD}{YX}$

Explanation:

The two rectangles are similar with similarity ratio 2.

Corresponding sides of similar rectangles are in proportion, so

$\frac{BC}{XY} = \frac{AB}{WX} = 2$ .

Since opposite sides of a rectangle are congruent, XY = WZ , so

$\frac{BC}{WZ}= 2$ .

$\overline{AC}$ and $\overline{WY}$ are diagonals of the rectangle. If they are constructed, then, since $\frac{AB}{WX} = \frac{BC}{XY}$ and $\angle B \cong \angle X$ (both are right angles), by the Side-Angle-Side Similarity Theorem, $\bigtriangleup ABC \cong \bigtriangleup WXY$ . By similarity, $\frac{AC}{WY} = \frac{AB}{WX} =2$ .

The ratio of the perimeters of the rectangles is

$\frac{2 \cdot AB + 2 \cdot BC }{2\cdot WX + 2 \cdot XY}$

$= \frac{ AB + BC }{ WX + XY}$ ,

It follows from $\frac{AB}{WX} =\frac{BC}{XY} = 2$ and a property of proportions that this ratio is equal to .

However,

$\frac{CD}{YX}$

is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.

Report an Error

Example Question #5 : Calculating Whether Rectangles Are Similar

A certain rectangle has a length of and a width of . Which of the following dimensions would another rectangle need in order for it to be similar?

Possible Answers:

L=12,W=5

L=32,W=12

L=24,W=10

L=18,W=7

L=20,W=8

Correct answer:

L=20,W=8

Explanation:

In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:

$Given:\frac{L}{W}=\frac{15}{6}=\frac{5}{2}$

So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:

$Option1:\frac{L}{W}=\frac{18}{7}$

$Option2:\frac{L}{W}=\frac{12}{5}$

$Option3:\frac{L}{W}=\frac{24}{10}=\frac{12}{5}$

$Option4:\frac{L}{W}=\frac{32}{12}=\frac{8}{3}$

$Option5:\frac{L}{W}=\frac{20}{8}=\frac{5}{2}$

A rectangle with a length of and a width of has the same ratio of dimensions as a rectangle with a length of and a width of , so these two rectangles are similar.

Report an Error

Example Question #6 : Calculating Whether Rectangles Are Similar

An engineer is making a scale model of a building. The real building needs to have a width of 54m and a length of 164m . If the engineer's scale model has a width of 0.05m , what does the length of the model need to be?

Possible Answers:

0.15m

0.35m

15.18m

0.92m

Correct answer:

0.15m

Explanation:

An engineer is making a scale model of a building. The real building needs to have a width of and a length of . If the engineer's scale model has a width of , what does the length of the model need to be?

To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.

So, we find a length that will make the model accurate, we need a ratio. Try the following:

$\frac{0.05}{54}=\frac{l_m}{164}$

$l_m=\frac{.05}{54}*164=0.15m$

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Houston, SAT Tutors in Houston, GMAT Tutors in Seattle, GRE Tutors in Boston, Statistics Tutors in Chicago, GMAT Tutors in Boston, Statistics Tutors in Philadelphia, ISEE Tutors in Phoenix, Reading Tutors in Seattle, Physics Tutors in Washington DC

Popular Courses & Classes

GRE Courses & Classes in Philadelphia, SAT Courses & Classes in Atlanta, SSAT Courses & Classes in Seattle, Spanish Courses & Classes in Los Angeles, MCAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Philadelphia, ACT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Phoenix, LSAT Courses & Classes in Chicago, Spanish Courses & Classes in Dallas Fort Worth

Popular Test Prep

MCAT Test Prep in Denver, SSAT Test Prep in Boston, GMAT Test Prep in Phoenix, SAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Boston, ISEE Test Prep in Denver, ACT Test Prep in Dallas Fort Worth, ISEE Test Prep in Dallas Fort Worth, MCAT Test Prep in Washington DC, SAT Test Prep in Houston

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : Calculating whether rectangles are similar

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating Whether Rectangles Are Similar

Example Question #2 : Calculating Whether Rectangles Are Similar

Example Question #3 : Calculating Whether Rectangles Are Similar

Example Question #4 : Calculating Whether Rectangles Are Similar

Example Question #5 : Calculating Whether Rectangles Are Similar

Example Question #6 : Calculating Whether Rectangles Are Similar

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details