GMAT Math : DSQ: Calculating the length of the side of an acute / obtuse triangle

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 #41 : Acute / Obtuse Triangles

Is the triangle isosceles?

Statement 1: The triangle has vertices A(1,5), B(4,2), and C(5,6).

Statement 2: $\overline{AB} = \sqrt{18}, \overline{AC} = \sqrt{17}, \overline{BC} = \sqrt{17}$

Possible Answers:

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

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

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

For a triangle to be isosceles, two of the sides must be equal. To determine wheter this is true, we must have the three side lengths. Statement 2 gives us those three side lengths. However, Statement 1 also gives us all of the information we need by giving us the three vertices. By using the distance formula, we can easily get the three triangle sides from the vertices. Therefore both statements alone are sufficient.

Report an Error

Example Question #1 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Rect

Note: Figure NOT drawn to scale.

The above shows a triangle $\Delta MCT$ inscribed inside a rectangle $\textrm{Rect} \; RECT$ . is $\Delta MCT$ isosceles?

Statement 1: is the midpoint of $\overline{RE}$ .

Statement 2: $\angle RTM \cong \angle ECM$

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:

We show Statement 1 alone is sufficient:

If is the midpoint of $\overline{RE}$ , then MR = MC . Opposite sides of a rectangle are congruent, so RT = EC ; all angles of a rectangle, being right angles, are congruent, so $\angle MRT \cong \angle MEC$ . This sets up the conditions for the Side-Angle-Side Theorem, and $\Delta MRT \cong \Delta MEC$ . Consequently, MT = MC , and $\Delta MCT$ is isosceles.

Now, we show Statement 2 alone is sufficient:

If $\angle RTM$ , and $\angle ECM$ are congruent, then $\angle MTC$ and $\angle MCT$ , being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, $\Delta MCT$ is isosceles.

Report an Error

Example Question #292 : Geometry

Which side of $\Delta ABC$ is the longest?

Statement 1: $\angle B$ is an obtuse angle.

Statement 2: $\angle A$ and $\angle C$ are both acute angles.

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 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:

If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.

If we know that $\angle B$ is an obtuse angle, however, we can deduce that $\angle A$ and $\angle C$ are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that $\angle B$ has the greatest measure, and that its opposite side, $\overline{AC}$ , is the longest.

Report an Error

Example Question #293 : Geometry

Is $\Delta ABC$ isosceles?

Statement 1: $\Delta ABC \sim \Delta DEF$

Statement 2: DE = 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.

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:

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

Explanation:

Statement 1 alone does not tell us anything unless we know the relative lengths of the sides of $\Delta DEF$ ; Statement 2 only gives us information about another triangle.

Suppose we assume both statements. Then by similarity,

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

Since DE = EF , then

$\frac{AB}{DE} \cdot DE = \frac{BC}{EF} \cdot EF$ , or

AB = BC .

This makes $\Delta ABC$ isosceles.

Report an Error

Example Question #294 : Geometry

Which of the three sides of $\Delta ABC$ is the longest?

Statement 1: $m \angle A =m \angle B + 20^{\circ }$

Statement 2: $m \angle C = m \angle A + 100^{\circ }$

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

Explanation:

The longest side of a triangle is opposite the angle of greatest measure.

From Statement 1 alone, we can find two possible scenarios with different answers:

Case 1:

$m \angle B = 20 ^{\circ }, m \angle A = 40 ^{\circ }, m \angle C = 120 ^{\circ }$

Case 2:

$m \angle C = 20 ^{\circ }, m \angle B = 70 ^{\circ }, m \angle A = 90 ^{\circ }$

In both cases, $m \angle A =m \angle B + 20^{\circ }$ , but in Case 1, $\overline{AB}$ is the longest side, and in Case 2, $\overline{BC }$ is the longest side.

From Statement 2 alone, however, we know that $m \angle C > 100^{\circ }$ , so $\angle C$ is obtuse and the other two angles are acute. That makes $\overline{AB}$ the longest side.

Report an Error

Example Question #295 : Geometry

True or false: $\Delta ABC$ is scalene.

Statement 1: $\angle A \ncong \angle B$

Statement 2: $\overline{AB} \ncong \overline{BC}$

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true.

By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1, $\overline{AC} \ncong \overline{BC}$ . Statement 2 alone establishes that $\overline{AB} \ncong \overline{BC}$ . However, the two statements together do not establish whether or not $\overline{AB} \ncong \overline{AC}$ , so it is not clear whether $\Delta ABC$ is scalene or isosceles.

Report an Error

Example Question #296 : Geometry

True or false: $\Delta ABC$ is scalene.

Statement 1: $\overline{AC} \ncong \overline{BC}$

Statement 2: $\angle B \cong \angle C$

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.

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.

Correct answer:

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

Explanation:

By definition, a scalene triangle has three noncongruent sides.

Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side $\overline{AB}$ is congruent to either of the other sides.

Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that $\overline{AB} \cong \overline{AC}$ . The triangle cannot be scalene.

Report an Error

Example Question #297 : Geometry

True or false: $\Delta ABC$ is scalene.

Statement 1: AB > AC

Statement 2: BC > AC

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 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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

By definition, a scalene triangle has three noncongruent sides.

If AB = 7, AC = 5, BC = 6 , then AB > AC and BC > AC , and the triangle is scalene.

If AB = 7, AC = 5, BC = 7 , then AB > AC and BC > AC , but AB = BC , so the triangle is not scalene.

The two statements together are insufficient.

Report an Error

All GMAT Math Resources

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

Popular Subjects

MCAT Tutors in Washington DC, GMAT Tutors in Los Angeles, Computer Science Tutors in Dallas Fort Worth, MCAT Tutors in Miami, SSAT Tutors in San Diego, Calculus Tutors in Dallas Fort Worth, French Tutors in Washington DC, Statistics Tutors in Phoenix, Calculus Tutors in Seattle, Chemistry Tutors in Washington DC

Popular Courses & Classes

LSAT Courses & Classes in Seattle, MCAT Courses & Classes in New York City, Spanish Courses & Classes in San Diego, ISEE Courses & Classes in Seattle, GMAT Courses & Classes in Boston, MCAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Philadelphia, SAT Courses & Classes in Los Angeles, ACT Courses & Classes in Houston, GMAT Courses & Classes in Philadelphia

Popular Test Prep

ACT Test Prep in San Diego, GRE Test Prep in Phoenix, ISEE Test Prep in Seattle, SAT Test Prep in Boston, ISEE Test Prep in Denver, GMAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Boston, ACT Test Prep in Houston, SAT Test Prep in New York City, SSAT Test Prep in Denver

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: Calculating the length of the side of an acute / obtuse triangle

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #41 : Acute / Obtuse Triangles

Example Question #1 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #292 : Geometry

Example Question #293 : Geometry

Example Question #294 : Geometry

Example Question #295 : Geometry

Example Question #296 : Geometry

Example Question #297 : Geometry

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details