GMAT Math : DSQ: Understanding rays

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 : Geometry

Lines

Note: Figure NOT drawn to scale.

Evaluate .

Statement 1: AD = 24

Statement 2: BE = 24

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Even with both statements, cannot be determined because the length of is missing.

For example, we can have AB = DE = 8 and BD=16 , making AE = 32 ; or, we can have AB = DE = 10 and BD=14 , making AE = 34 . Neither scenario violates the conditions given.

Report an Error

Example Question #2 : Geometry

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Statement 1: is on $\overleftrightarrow{AC}$

Statement 2: is on $\overleftrightarrow{AB}$

Possible Answers:

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.

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.

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

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements are equivalent, as both are equivalent to stating that , , and are collinear. Therefore, it suffices to determine whether the fact that the points are collinear is sufficient to answer the question.

Rays

In both of the above figures, , , and are collinear, so the conditions of both statements are met. But in the top figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since is on $\overrightarrow{AB}$ ; in the bottom figure, since and are on opposite sides of , $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Report an Error

Example Question #3 : Geometry

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Statement 1: $AC = 2 \cdot AB$ .

Statement 2: is the midpoint of $\overline{AC}$ .

Possible Answers:

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.

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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:

We show Statement 1 alone is insufficient to determine whether the two rays are the same by looking at the figures below. In the first figure, is the midpoint of $\overline{AC}$ .

Rays

In both figures, $AC = 2 \cdot AB$ . But only in the second figure, and are on the opposite side of the line from , so only in the second figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Assume Statement 2 alone. If is the midpoint of $\overline{AC}$ , then, as seen in the top figure, is on $\overrightarrow{AC}$ . Therefore, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, not opposite rays.

Report an Error

Example Question #4 : Geometry

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays.

Statement 1: AB+BC > AC

Statement 2: AB+ AC > BC

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.

BOTH statements TOGETHER are insufficient 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.

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 alone does not answer the question.

Case 1: Examine the figure below.

Rays

AB+BC= AB + AB + AC > AC ,

thereby meeting the condition of Statement 1.

Also, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays, since and are on opposite sides of the same line from .

Case 2: Suppose , , and are noncollinear.

The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

AB+BC > AC .

Furthermore, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are not part of the same line and are not opposite rays.

Now assume Statement 2 alone. As can be seen in the diagram above, if $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are opposite rays, then by segment addition, AB+BC = AC , making Statement 2 false. Contrapositively, if Statement 2 holds, and $AB+ AC \ne BC$ , then $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are not opposite rays.

Report an Error

Example Question #1 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: $AB = 2 \cdot AC$ .

Statement 2: is the midpoint of $\overline{AB}$ .

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.

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 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:

We show Statement 1 alone is insufficient to determine whether the two rays are the same by looking at the figures below:

Rays

In both figures, $AB = 2 \cdot AC$ , but only in the first figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Assume Statement 2 alone. If is the midpoint of $\overline{AB}$ , must be on $\overrightarrow{AB}$ , as in the top figure, so $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are one and the same.

Report an Error

Example Question #121 : Data Sufficiency Questions

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: AB+BC > AC

Statement 2: AB+ AC > BC .

Possible Answers:

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We show that both statements together provide insufficient information by giving two scenarios in which both statements are true.

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the Triangle Inequality Theorem,

AB+BC > AC and

AB + AC > BC .

Also, since the three points are not on a single line, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are parts of different lines and cannot be the same ray.

Case 2: $\overline{AB}$ with length 2 and midpoint .

Rays

AB+BC =2+1 = 3 and AC = 1 , so AB+BC > AC ; similarly, AB + AC > BC . Also, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since they have the same endpoint and is on $\overrightarrow{AB}$ .

Report an Error

Example Question #3 : Dsq: Understanding Rays

, , and are distinct points.

True or false: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Statement 1: , , and are collinear.

Statement 2: AB + BC = AC .

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 insufficient 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 sufficient to answer the question, but NEITHER 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:

Statement 1 alone does not prove the rays to be the same or different, as seen in these diagrams:

Rays

In both figures, , , and are collinear, satisfying the condition of Statement 1. But In the top figure, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray, since is on $\overrightarrow{AB}$ ; in the bottom figure, since is not on $\overrightarrow{AB}$ , $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are distinct rays.

Assume Statement 2 alone. Suppose $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are not the same ray. Then one of two things happens:

Case 1: , , and are noncollinear. The three points are vertices of a triangle, and by the triangle inequality,

AB + BC > AC ,

contradicting Statement 2.

Case 2: , , and are collinear. must be between and , as in the bottom figure, since if it were not, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ would be the same ray. By segment addition,

AB + AC = BC

AB + AC+AB = BC+AB

$AB + BC = AC + 2 \cdot AB > AC$ ,

contradicting Statement 2.

By contradiction, $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray.

Report an Error

All GMAT Math Resources

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

Popular Subjects

ISEE Tutors in Chicago, Biology Tutors in San Francisco-Bay Area, Algebra Tutors in Los Angeles, SAT Tutors in Seattle, Physics Tutors in Phoenix, LSAT Tutors in San Francisco-Bay Area, MCAT Tutors in Los Angeles, French Tutors in San Diego, SSAT Tutors in Seattle, Biology Tutors in Seattle

Popular Courses & Classes

GRE Courses & Classes in Chicago, SSAT Courses & Classes in Denver, LSAT Courses & Classes in New York City, ISEE Courses & Classes in Miami, GRE Courses & Classes in Houston, GRE Courses & Classes in Denver, SAT Courses & Classes in Houston, LSAT Courses & Classes in Miami, Spanish Courses & Classes in Denver, SSAT Courses & Classes in New York City

Popular Test Prep

ACT Test Prep in Los Angeles, SSAT Test Prep in Los Angeles, GRE Test Prep in Chicago, MCAT Test Prep in New York City, ISEE Test Prep in San Diego, GMAT Test Prep in Chicago, SAT Test Prep in Houston, ACT Test Prep in Houston, SSAT Test Prep in Seattle, ISEE Test Prep in Chicago

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 : DSQ: Understanding rays

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Geometry

Example Question #2 : Geometry

Example Question #3 : Geometry

Example Question #4 : Geometry

Example Question #1 : Dsq: Understanding Rays

Example Question #121 : Data Sufficiency Questions

Example Question #3 : Dsq: Understanding Rays

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details