GMAT Math : Triangles

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 #5 : Dsq: Calculating Whether Right Triangles Are Congruent

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB = DF

Statement 2: BC = DF

Possible Answers:

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

In any right triangle, the hypotenuse must have length greater than either leg. Therefore, DF > DE and DF > EF .

Assume Statement 1 alone. For $\bigtriangleup ABC \cong \bigtriangleup DEF$ , it must hold that AB = DE . However, AB = DF and DF > DE , so AB > DE . The statement proves that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is false. By a similar argument, Statement 2 proves $\bigtriangleup ABC \cong \bigtriangleup DEF$ is false.

Report an Error

Example Question #8 : Dsq: Calculating Whether Right Triangles Are Congruent

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: $\angle A$ and $\angle F$ are complementary angles.

Statement 2: AB + BC < DE + EF

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.

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 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:

Assume Statement 1 alone. In any right triangle, the two acute angles are complementary. Therefore, $\angle A$ and $\angle C$ are complementary angles, as are $\angle D$ and $\angle F$ are complementary angles. Also, two angles complementary to the same angle are coongruent, so, since $\angle A$ and $\angle F$ are complementary angles, $\angle A \cong \angle D$ and $\angle C \cong \angle F$ . From the congruences of all three pairs of corresponding angles, it follows that $\bigtriangleup ABC$ and $\bigtriangleup DEF$ are similar, but without any side comparisons, congruence between the triangles cannot be proved or disproved.

Assume Statement 2 alone. If $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then corresponding sides are congruent, so AB = DE and BC = EF . Therefore, AB + BC = DE + EF . But Statement 2 tells us that this is false. Therefore, we can determine that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is a false statement.

Report an Error

Example Question #4 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB = BC

Statement 2: $DF = DE \sqrt {2}$

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true. Statement 1 establishes that $\bigtriangleup ABC$ has two congruent legs, making it a 45-45-90 triangle. Statement 2 establishes that $\bigtriangleup DEF$ has a hypotenuse that has length $\sqrt{ 2}$ times that of a leg, making it also a 45-45-90 triangle. The triangles have the same angle measures, so they are similar by the Angle-Angle Postulate. However, we are not given any actual lengths or any relationship between the lengths of the corresponding sides of different triangles, so we cannot determine whether the triangles are congruent or not.

Report an Error

Example Question #11 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: $\left (DE \right ) ^{2} +(EF) ^{2} = \left ( AC\right )^{2}$

Statement 2: $\left ( AC \right )^{2} - \left ( AB\right )^{2} > \left ( EF\right )^{2}$

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:

Assume Statement 1 only. By the Pythagorean Theorem, $\left (DE \right ) ^{2} +(EF) ^{2} = \left ( DF\right )^{2}$ , so $\left ( AC\right )^{2}= \left ( DF\right )^{2}$ ; subsequently, AC= DF , and one side congruence is proved. However, this, along with one angle congruence - the congruence of right angles $\angle B$ and $\angle E$ - is not enough to prove or disprove triangle congruence.

Assume Statement 2 only. By the Pythagorean Theorem, $\left ( AC \right )^{2} - \left ( AB\right )^{2} = \left ( BC\right )^{2}$ . Since $\left ( AC \right )^{2} - \left ( AB\right )^{2} > \left ( EF\right )^{2}$ , it follows that $\left ( BC \right )^{2} > \left ( EF\right )^{2}$ , and, subsequently, BC > EF . Since, if $\bigtriangleup ABC \cong \bigtriangleup DEF$ , BC = EF , it follows by contradiction that $\bigtriangleup ABC \cong \bigtriangleup DEF$ is a false statement.

Report an Error

Example Question #12 : Dsq: Calculating Whether Right Triangles Are Congruent

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: A single circle can be constructed that passes through A,B,C,D,E, and .

Statement 2: AB = 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.

BOTH statements TOGETHER are insufficient 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.

Correct answer:

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

Explanation:

Assume Statement 1 alone is true. Since each right triangle is inscribed inside a circle, each of the right angles is inscribed in that circle, and each intercepts a semicircle. That makes the two hypotenuses of the triangles, $\overline{AC}$ and $\overline{DF}$ , diameters; since they are on the same circle, their lengths are the same. However, no information is given about any of the other sides or angles, so no congruence can be proved or disproved.

Statement 2 alone gives us only congruence between one set of corresponding legs, which, along with one angle congruence, is insufficient to prove or to disprove triangler congruence.

Now assume both statements are true. From Statement 1, it follows that congruence of hypotenuses, and Statement 2 gives us congruence of corresponding legs; this sets up the conditions of the Hypotenuse-Leg Theorem, so it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Report an Error

Example Question #12 : Dsq: Calculating Whether Right Triangles Are Congruent

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ have the same area.

Statement 2: AB + BC = DE+EF

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:

Assume both statements are true. We show that both statements are insufficient to answer the question.

Suppose AB = DE = 6 and BC = EF= 8 .

AB + BC = 6+8 = 14 and DE+EF = 6 + 8 = 14 , satisfying the conditions of Statement 2.

The area of a right triangle is half the product of the lengths of its legs. Each triangle therefore has as its area $\frac{1}{2} \cdot 6 \cdot 8 = 24$ , making the areas the same; Statement 1 is satisfied.

Since corresponding legs of the triangles are congruent, $\bigtriangleup ABC \cong \bigtriangleup DEF$ by the Side-Angle-Side Postulate.

Now, suppose AB = EF= 6 and BC = DE= 8 .

AB + BC = 6+8 = 14 and DE+EF = 8+6 = 14 , satisfying the conditions of Statement 2.

However, $AB \ne DE$ and $BC \ne EF$ , so it is not true that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Report an Error

Example Question #111 : Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , with right angles $\angle B, \angle E$

True or false: $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Statement 1: AB = BC = DE= EF

Statement 2: $m \angle A = m \angle C = m \angle D = m \angle F$

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 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.

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. The statement gives that all four legs of both triangles are congruent - specifically, AB = DE and BC = EF . Since the right angles are also congruent, then, by the Side-Angle-Side Postulate, $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Assume Statement 2 alone. The statement gives that all four acute angles are congruent - specifically, that $m \angle A = m \angle D$ and $m \angle C = m \angle F$ . However, since we do not have any congruence or noncongruence between corresponding sides, congruence of the triangles cannot be proved or disproved.

Report an Error

Example Question #112 : Triangles

Two ships left New York at the same time. One ship has been moving due east the entire time at a speed of 50 nautical miles per hour. How far apart are the ships now?

1) The other ship has been moving due south the entire time.

2) The other ship has been moving at a rate of 60 nautical miles per hour the entire time.

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 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.

Correct answer: BOTH statements TOGETHER are not sufficient to answer the question.

Explanation:

In order to determine how far apart the ships are now, it is necessary to know how far each ship is from New York now. Even knowing both statements, however, you only know that the paths are at right angles, and that the ratio of the two distances is 6 to 5. Without knowing the time elapsed, which is not given, you cannot tell how far apart the ships are.

The answer is that both statements together are insufficent to answer the question.

Report an Error

Example Question #2 : Dsq: Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Right triangle DEF is similar to triangle HJK .

Find the hypotenuse of triangle DEF

I) Triangle HJK has side lengths of 12/16/20 .

II) The shortest side of DEF is .

Possible Answers:

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

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

Each statement alone is enough to solve the question.

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

Both statements taken together are sufficient to solve the question.

Correct answer:

Both statements taken together are sufficient to solve the question.

Explanation:

We need both statements here because we need to work with ratios.

If you notice that a 12/16/20 triangle follows the 3/4/5 triangle ratio, then it is even easier. Otherwise, you can set up the following proportion and solve to find the answer.

$\small \frac{12}{24}=\frac{20}x$

Where x is the hypotenuse of triangle HJK.

Report an Error

Example Question #1 : Dsq: Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Find the hypotenuse of $\Delta HLC$ .

I) The shortest side is half of the second shortest side.

II) The middle side is fathoms long.

Possible Answers:

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

Both statements are needed to answer the question.

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.

Correct answer:

Both statements are needed to answer the question.

Explanation:

To find the hypotenuse of a right triangle we can use the classic Pythagorean Theorem. To do so, however, we need to know the other two sides.

I) Tells us how to relate the other two sides to eachother.

$SS=\frac{1}{2}MS$

II) Gives us the length of one side.

MS=13

Use II) and I) to find the two short sides. Then use Pythagorean Theorem to find the perimeter.

$SS=\frac{1}{2}\cdot 13=\frac{13}{2}$

SS^2+MS^2=H^2 , where SS represents the shortest side, MS represents the middle side, and H represents the hypotenuse.

$\left(\frac{13}{2} \right )^2+13^2=211.25 \rightarrow H=14.5$

$P=\frac{13}{2}+13+14.5$

P=34

Report an Error

All GMAT Math Resources

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

Popular Subjects

French Tutors in Boston, Physics Tutors in Dallas Fort Worth, French Tutors in Los Angeles, MCAT Tutors in Miami, MCAT Tutors in Philadelphia, Biology Tutors in San Francisco-Bay Area, English Tutors in Washington DC, Physics Tutors in Miami, ISEE Tutors in Washington DC, ISEE Tutors in Atlanta

Popular Courses & Classes

LSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Seattle, MCAT Courses & Classes in Houston, SSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Philadelphia, Spanish Courses & Classes in Denver, ACT Courses & Classes in Denver, SSAT Courses & Classes in Phoenix, SAT Courses & Classes in Houston, MCAT Courses & Classes in San Francisco-Bay Area

Popular Test Prep

ACT Test Prep in Seattle, MCAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in New York City, GMAT Test Prep in Dallas Fort Worth, ACT Test Prep in Denver, ACT Test Prep in Dallas Fort Worth, LSAT Test Prep in Washington DC, GMAT Test Prep in Chicago, MCAT Test Prep in Houston, MCAT Test Prep in Miami

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 : Triangles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Dsq: Calculating Whether Right Triangles Are Congruent

Example Question #8 : Dsq: Calculating Whether Right Triangles Are Congruent

Example Question #4 : Right Triangles

Example Question #11 : Right Triangles

Example Question #12 : Dsq: Calculating Whether Right Triangles Are Congruent

Example Question #12 : Dsq: Calculating Whether Right Triangles Are Congruent

Example Question #111 : Triangles

Example Question #112 : Triangles

Example Question #2 : Dsq: Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #1 : Dsq: Calculating The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details