GMAT Math : Circles

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 The Ratio Of Diameter And Circumference

What is the circumference of the circle?

The diameter of the circle is .
The area of the circle is $64\pi$ .

Possible Answers:

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

Each statement alone is sufficient to answer the question.

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

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Correct answer:

Each statement alone is sufficient to answer the question.

Explanation:

Statement 1: We can calculate the circumference using the given diameter.

$circumference=d\pi =16\pi$

Statement 2: To find the circumference, we must first find the radius of the circle using the given area.

$64\pi =r^2\pi$

r^2=64

r=8

We can plug this value into the equation for circumference:

$C=2r\pi=16\pi$

Each statement alone is sufficient to answer the question.

Report an Error

Example Question #1 : Chords

Chords

Note: Figure NOT drawn to scale

Refer to the above figure. Is $\Delta ABC$ an isosceles triangle?

Statement 1: $\widehat{ABC}$ and $\widehat{BCA}$ have equal length.

Statement 2: $\widehat{ABC}$ and $\widehat{BCA}$ have equal degree measure.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 and Statement 2 are equivalent, as two arcs on the same circle have the same length if and only if they have the same degree measure. We only need to prove the sufficiency or insufficiency of one statement to answer the question.

Choose Statement 2. If $\widehat{ABC}$ and $\widehat{BCA}$ have equal degree measure, then their minor arcs $\widehat{AC}$ and $\widehat{BA}$ do also. Congruent arcs on the same circle have congruent chords, so $\overline{AC} \cong \overline{ BA}$ , and this proves $\Delta ABC$ isosceles.

Report an Error

Example Question #1 : Chords

Chord

Note: Figure NOT drawn to scale.

Give the length of chord $\overline{AB}$ .

Statement 1: Minor arc $\widehat{AB}$ has length $8 \pi$ .

Statement 2: Major arc $\widehat{ACB}$ has length $16 \pi$ .

Possible Answers:

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.

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.

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 is insufficient to give the length of the chord, since no other information is known about the major arc, the circle, or the angle. For similar reasons, Statement 2 alone is insufficient.

If both statements are assumed, then it is possible to add the arc lengths to get the circumference of the circle, which is $8 \pi + 16 \pi = 24 \pi$ . It follows that the radius is $24 \pi \div 2 \pi = 12$ , and that $n =\frac{ 8 \pi }{24 \pi } \times 360 ^{\circ } = 120 ^{\circ }$ . From this information, can be calculated by bisecting the triangle into two 30-60-90 triangles with a perpendicular bisector from , and applying the 30-60-90 theorem.

Report an Error

Example Question #512 : Geometry

Chord

Note: Figure NOT drawn to scale.

In the above figure, is the center of the circle, and $\Delta OAB$ is equilateral. Give the length of Give the length of chord $\overline{AB}$ .

Statement 1: The circle has area $64 \pi$ .

Statement 2: $\Delta OAB$ has perimeter .

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Since $\Delta OAB$ is equilateral, the length of chord $\overline{AB}$ is equivalent to the length of $\overline{OA}$ , and, subsequently, the radius of the circle. If Statement 1 alone is assumed, the radius of the circle can be calculated using the area formula.

If Statement 2 alone is assumed, the length of $\overline{AB}$ is one third of the known perimeter.

Report an Error

Example Question #4 : Dsq: Calculating The Length Of A Chord

Chord

Note: Figure NOT drawn to scale.

Examine the above figure. True or false: $\overline{AB} \cong \overline{CD}$ .

Statement 1: Arc $\widehat{CAB}$ is longer than arc $\widehat{ACD}$ .

Statement 2: Arc $\widehat{ABD}$ is longer than arc $\widehat{CDB}$ .

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

For two chords in the same circle to be congruent, it is necessary and sufficient that their arcs have the same length.

By arc addition, the length of $\widehat{CAB}$ is the sum of the lengths of $\widehat{CA }$ and $\widehat{ AB}$ , which we will call and , respectively. Similarly, the length of $\widehat{ACD}$ is the sum of the lengths of $\widehat{AC}$ and $\widehat{ CD}$ , which we will call and , respectively.

If Statement 1 alone is assumed,

x+y > x+ z

Subsequently,

y > z ,

so $\widehat{ AB}$ is longer than $\widehat{ CD}$ . The arcs are of unequal length so their chords are as well. This makes Statement 1 sufficient to answer the question. A similar argument can be made that Statement 2 alone answers the question.

Report an Error

Example Question #1 : Dsq: Calculating The Length Of A Chord

Chord

Note: Figure NOT drawn to scale

Examine the above figure. True or false: $\overline{AB} \cong \overline{YZ}$ .

Statement 1: $\overline{XY} \cong \overline{BC}$

Statement 2: $\angle ABC \cong \angle XYZ$

Possible Answers:

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.

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:

Statement 1 only gives information about two other chords, whose relationship with the first two is not known. Statement 2 only gives the congruence of two inscribed angles - and, subsequently, since congruent inscribed angles intercept congruent arcs, that $m \widehat{AC} = m \widehat{ZX}$ - but gives no information about the individual sides.

Assume both statements.

Congruent chords of the same circle must have arcs of the same degree measure, so, from Statement 1, since $\overline{XY} \cong \overline{BC}$ , then $m \widehat{CB} = m \widehat{XY}$ . From Statement 2, as stated before, $m \widehat{AC} = m \widehat{ZX}$ . Then,

$m \widehat{AC} + m \widehat{CB} = m \widehat{ZX} + m \widehat{XY}$

By arc addition, this statement becomes

$m \widehat{AB} = m \widehat{ZY}$ .

Since congruent chords on the same circle have congruent arcs,

$\overline{AB} \cong \overline{YZ}$ .

Report an Error

Example Question #515 : Geometry

Chord

Note: Figure NOT drawn to scale

Examine the above figure. True or false: $\overline{AB} \cong \overline{YZ}$ .

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

Statement 2: $\angle ABC \ncong \angle XYZ$

Possible Answers:

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.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Congruent chords of the same circle must have arcs of the same degree measure, so

$\overline{AB} \cong \overline{YZ}$ if and only if $m \widehat{AB} = m \widehat{ZY}$ .

Assume both statements. Then , since, in the same circle, congruent arcs have congruent chords, it follows from Statement 1 that

$m \widehat{BC} \ne m \widehat{XY}$ .

Also, since congruent inscribed angles intercept congruent arcs, it follows from Statement 2 that

$m \widehat{AC} \ne m \widehat{XZ}$

By arc addition,

$m \widehat{AB} = m \widehat{ZY}$

can be expressed as

$m \widehat{AC} +m \widehat{BC} = m \widehat{XZ} + m \widehat{XY}$ .

Examples of the values of the four arc measures $m \widehat{AC}$ , $m \widehat{XZ}$ , $m \widehat{BC}$ , and $m \widehat{XY}$ can easily be found to make $m \widehat{AC} \ne m \widehat{XZ}$ and $m \widehat{BC} \ne m \widehat{XY}$ so that

$m \widehat{AC} +m \widehat{BC} = m \widehat{XZ} + m \widehat{XY}$ is either true or false; consequently, $\overline{AB} \cong \overline{YZ}$ may be true or false.

The two statements together are insufficient.

Report an Error

Example Question #2 : Dsq: Calculating The Length Of A Chord

Chord

Note: Figure NOT drawn to scale.

Examine the above diagram. True or false: $\overline{AB} \cong \overline{CD}$ .

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

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

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If two chords of a circle intersect inside it, and two more chords are constructed connecting endpoints, as is the case here, the resulting triangles are similar - that is,

$\Delta ABM \sim \Delta DCM$

$\overline{AB} \cong \overline{CD}$ if and only if the triangles are congruent. From either statement alone, we are given a side congruence - from Statement 1 alone it follows that $\overline{BM} \cong \overline{DM}$ , and from Statement 2 alone, it follows that $\overline{AM} \cong \overline{CM}$ . Either way, the resulting side congruency, along with two angle congruencies following from the similarity of the triangles, prove by way of the Angle-Sude-Angle Postulate that $\Delta ABM \cong \Delta DCM$ , and, subsequently, that $\overline{AB} \cong \overline{CD}$ .

Report an Error

Example Question #3 : Dsq: Calculating The Length Of A Chord

Chord

Note: Figure NOT drawn to scale.

Examine the above figure. True or false: $\overline{AB} \cong \overline{CD}$ .

Statement 1: Arcs $\widehat{ AB}$ and $\widehat{ CD}$ have the same length.

Statement 2: Arcs $\widehat{ADB}$ and $\widehat{CBD}$ have the same degree measure.

Possible Answers:

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

For two chords in the same circle to be congruent, it is necessary and sufficient that their minor arcs have the same length. Statement 1 asserts this, so it is sufficient to answer the question.

It is also necessary and sufficient that their major arcs have the same degree measure. Statement 2 alone asserts this, so it is sufficient to answer the question.

Report an Error

All GMAT Math Resources

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

Popular Subjects

Reading Tutors in New York City, Chemistry Tutors in New York City, English Tutors in Miami, French Tutors in Denver, Spanish Tutors in San Diego, English Tutors in New York City, GMAT Tutors in Chicago, Chemistry Tutors in Dallas Fort Worth, Statistics Tutors in Boston, Spanish Tutors in Denver

Popular Courses & Classes

SSAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Atlanta, SAT Courses & Classes in Houston, Spanish Courses & Classes in Seattle, Spanish Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Atlanta, SSAT Courses & Classes in Chicago, MCAT Courses & Classes in Miami, Spanish Courses & Classes in San Diego, SAT Courses & Classes in Boston

Popular Test Prep

MCAT Test Prep in San Diego, GRE Test Prep in Chicago, SAT Test Prep in Houston, GMAT Test Prep in Atlanta, LSAT Test Prep in Houston, SSAT Test Prep in Boston, LSAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in San Francisco-Bay Area, GRE Test Prep in Philadelphia, ACT 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 : Circles

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #5 : Dsq: Calculating The Ratio Of Diameter And Circumference

Example Question #1 : Chords

Example Question #1 : Chords

Example Question #512 : Geometry

Example Question #4 : Dsq: Calculating The Length Of A Chord

Example Question #1 : Dsq: Calculating The Length Of A Chord

Example Question #515 : Geometry

Example Question #2 : Dsq: Calculating The Length Of A Chord

Example Question #3 : Dsq: Calculating The Length Of A Chord

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details