Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Math : How to find the length of a chord

Study concepts, example questions & explanations for ISEE Upper Level Math

Create An Account

All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Find The Length Of A Chord

Which equation is the formula for chord length?

Note: is the radius of the circle, and is the angle cut by the chord.

Possible Answers:

2rsinC

$rsin(\frac{C}{2})$

rsinC

$4rsin(\frac{C}{2})$

$2rsin(\frac{C}{2})$

Correct answer:

$2rsin(\frac{C}{2})$

Explanation:

The length of a chord of a circle is calculated as follows:

Chord length = $2rsin(\frac{C}{2})$

Report an Error

Example Question #151 : Plane Geometry

The radius of a circle is $5\ cm$ , and the perpendicular distance from a chord to the circle center is $4\ cm$ . Give the chord length.

Possible Answers:

$8\ cm$

$6\ cm$

$7\ cm$

$6.5\ cm$

$7.5\ cm$

Correct answer:

$6\ cm$

Explanation:

Chord length = $2\sqrt{r^2-d^2}$ , where is the radius of the circle and is the perpendicular distance from the chord to the circle center.

Chord length = $2\sqrt{r^2-d^2}=2\sqrt{5^2-4^2}$

$=2\sqrt{25-16}=2\sqrt{9}=2\times 3$

$\Rightarrow$ Chord length = $6\ cm$

Report an Error

Example Question #152 : Plane Geometry

In the circle below, the radius is $5\ cm$ and the chord length is $8\ cm$ . Give the perpendicular distance from the chord to the circle center (d).

Chord

Possible Answers:

$3\ cm$

$4.7\ cm$

$5\ cm$

$3.5\ cm$

$4.5\ cm$

Correct answer:

$3\ cm$

Explanation:

Chord length = $2\sqrt{r^2-d^2}$ , where is the radius of the circle and is the perpendicular distance from the chord to the circle center.

Chord length = $2\sqrt{r^2-d^2}\Rightarrow 8=2\sqrt{5^2-d^2}\Rightarrow 4=\sqrt{5^2-d^2}$

$\Rightarrow 16=25-d^2\Rightarrow d^2=25-16=9$

$\Rightarrow d=3\ cm$

Report an Error

Example Question #153 : Plane Geometry

Give the length of the chord of a $90^{\circ }$ central angle of a circle with radius 18.

Possible Answers:

$9\sqrt{3}$

$18\sqrt{3}$

$18\sqrt{2}$

$9\sqrt{2}$

Correct answer:

$18\sqrt{2}$

Explanation:

The figure below shows $\angle AOB$ , which matches this description, along with its chord $\overline{AB}$ :

Chord

By way of the Isoscelese Triangle Theorem, $\Delta AOB$ can be proved a 45-45-90 triangle with legs of length 18, so its hypotenuse - the desired chord length - is $\sqrt{2}$ times this, or $18\sqrt{2}$ .

Report an Error

Example Question #151 : Plane Geometry

A $60^{\circ}$ central angle of a circle intercepts an arc of length $7 \pi$ ; it also has a chord. What is the length of that chord?

Possible Answers:

$14\sqrt{3}$

$\frac{21\sqrt{2}}{2}$

$14\sqrt{2}$

$7\sqrt{3}$

Correct answer:

Explanation:

The arc intercepted by a $60^{\circ}$ central angle is $\frac{60}{360}= \frac{1}{6}$ of the circle, so the circumference of the circle is $7 \pi \cdot 6 = 42\pi$ . The radius is the circumference divided by $2 \pi$ , or $42 \pi \div 2 \pi = 21$ .

The figure below shows a $60^{\circ}$ central angle $\angle AOB$ , along with its chord $\overline{AB}$ :

Chord

By way of the Isoscelese Triangle Theorem, $\Delta AOB$ can be proved equilateral, so AB = OA = 21 .

Report an Error

Example Question #154 : Plane Geometry

A $120^{\circ }$ central angle of a circle intercepts an arc of length $\frac{4 \pi}{3}$ ; it also has a chord. What is the length of that chord?

Possible Answers:

$2 \sqrt{2}$

$2 \sqrt{3}$

$\sqrt{2}$

$\sqrt{3}$

Correct answer:

$2 \sqrt{3}$

Explanation:

The arc intercepted by a $120^{\circ}$ central angle is $\frac{120}{360}= \frac{1}{3}$ of the circle, so the circumference of the circle is $\frac{4 \pi}{3}\cdot 3= 4\pi$ . The radius is the circumference divided by $2 \pi$ , or $4\pi \div 2 \pi = 2$ .

The figure below shows a $120^{\circ}$ central angle $\angle AOB$ , along with its chord $\overline{AB}$ and triangle bisector $\overline{OM}$ .

Chord

We will concentrate on $\Delta AOM$ , which is a 30-60-90 triangle. By the 30-60-90 Theorem,

$OM = \frac{1}{2} AO = \frac{1}{2} \cdot 2 = 1$

and

$AM = OM \cdot \sqrt{3}= 1\sqrt{3}= \sqrt{3}$

is the midpoint of $\overline{AB}$ , so

$AB = 2 \cdot AM = 2 \sqrt{3}$

Report an Error

Example Question #152 : Plane Geometry

Give the length of the chord of a $60^{\circ }$ central angle of a circle with radius 20.

Possible Answers:

$10\sqrt{3}$

The correct answer is not among the other choices.

$20\sqrt{2}$

$10\sqrt{2}$

$20\sqrt{3}$

Correct answer:

The correct answer is not among the other choices.

Explanation:

The figure below shows $\angle AOB$ , which matches this description, along with its chord $\overline{AB}$ :

Chord

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved equilateral, so AB = OA = 20 .

This answer is not among the choices given.

Report an Error

Example Question #153 : Plane Geometry

Give the length of the chord of a $120^{\circ }$ central angle of a circle with radius .

Possible Answers:

$12 \sqrt{2}$

$18 \sqrt{2}$

$12 \sqrt{3}$

Correct answer:

$12 \sqrt{3}$

Explanation:

The figure below shows $\angle AOB$ , which matches this description, along with its chord $\overline{AB}$ and triangle bisector $\overline{OM}$ .

Chord

We will concentrate on $\Delta AOM$ , which is a 30-60-90 triangle. By the 30-60-90 Theorem,

$OM = \frac{1}{2} AO = \frac{1}{2} \cdot 12 = 6$

and

$AM = OM \cdot \sqrt{3}= 6\sqrt{3}$

is the midpoint of $\overline{AB}$ , so

$AB = 2 \cdot AM = 2 \cdot 6 \sqrt{3}= 12 \sqrt{3}$

Report an Error

Example Question #152 : Plane Geometry

Chords

Figure NOT drawn to scale

In the figure above, evaluate .

Possible Answers:

t = 18

t= 20

t = 16

t= 12

Correct answer:

t= 12

Explanation:

If two chords of a circle intersect inside the circle, the product of the lengths of the parts of each chord is the same. In other words,

40 t = 20 (t+12)

Solving for - distribute:

40 t = 20 (t+12)

$40 t = 20 t+ 20 \cdot 12$

40 t = 20 t+ 240

Subtract 20t from both sides:

40 t- 20 t = 20 t+ 240 - 20 t

20 t = 240

Divide both sides by 20:

$20 t \div 20 = 240 \div 20$

t = 12

Report an Error

Example Question #6 : How To Find The Length Of A Chord

Secant

In the above figure, $\overline{NT}$ is a tangent to the circle.

Evaluate .

Possible Answers:

$6 \sqrt{10}$

$6\sqrt{6}$

Correct answer:

$6 \sqrt{10}$

Explanation:

If a secant segment and a tangent segment are constructed to a circle from a point outside it, the square of the distance to the circle along the tangent is equal to the product of the distances to the two points on the circle along the secant; in other words,

$t^{2} = 12 \cdot (12+ 18) = 12 \cdot 30 = 360$

Solving for :

$t^{2} = 360$

$t = \sqrt{360}$

Simplifying the radical using the Product of Radicals Principle, and noting that 36 is the greatest perfect square factor of 360:

$t = \sqrt{36} \cdot \sqrt{10} = 6 \sqrt{10}$

Report an Error

View Tutors

Gabby
Certified Tutor

University of Oregon, Bachelor in Arts, Journalism. University of Oregon, Current Grad Student, Communication, General.

View Tutors

Eva
Certified Tutor

Technicka Univerzita Liberec, Bachelor of Education, Education.

View Tutors

Jaleesa
Certified Tutor

Pomona College, Bachelors, Africana Studies.

All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

MCAT Tutors in Boston, Spanish Tutors in Phoenix, French Tutors in Washington DC, Physics Tutors in New York City, GMAT Tutors in San Francisco-Bay Area, Spanish Tutors in Dallas Fort Worth, Spanish Tutors in San Diego, English Tutors in Seattle, Spanish Tutors in Denver, Chemistry Tutors in Atlanta

Popular Courses & Classes

GRE Courses & Classes in Washington DC, ACT Courses & Classes in Atlanta, Spanish Courses & Classes in Los Angeles, MCAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Chicago, Spanish Courses & Classes in Boston, SAT Courses & Classes in Atlanta, SSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Philadelphia, ACT Courses & Classes in Washington DC

Popular Test Prep

SAT Test Prep in Miami, GMAT Test Prep in Seattle, ACT Test Prep in Atlanta, SAT Test Prep in Los Angeles, SSAT Test Prep in Washington DC, SSAT Test Prep in Philadelphia, LSAT Test Prep in Atlanta, ACT Test Prep in Houston, GMAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Los Angeles

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)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

ISEE Upper Level Math : How to find the length of a chord

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Find The Length Of A Chord

Example Question #151 : Plane Geometry

Example Question #152 : Plane Geometry

Example Question #153 : Plane Geometry

Example Question #151 : Plane Geometry

Example Question #154 : Plane Geometry

Example Question #152 : Plane Geometry

Example Question #153 : Plane Geometry

Example Question #152 : Plane Geometry

Example Question #6 : How To Find The Length Of A Chord

All ISEE Upper Level Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: