Call Now to Set Up Tutoring:

(888) 888-0446

SAT Math : Circles

Study concepts, example questions & explanations for SAT Math

Create An Account

All SAT Math Resources

16 Diagnostic Tests 660 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Circles

Two chords of a circle, $\overline{AB}$ and $\overline{CD}$ , intersect at a point . $\overline{CX}$ is twice as long as $\overline{AX}$ , BX = 20 , and DX = 10 .

Give the length of $\overline{AX}$ .

Possible Answers:

Insufficient information is given to find the length of $\overline{AX}$ .

$10 \sqrt{3}$

$10 \sqrt{2}$

$20\sqrt{2}$

Correct answer:

Insufficient information is given to find the length of $\overline{AX}$ .

Explanation:

Let stand for the length of $\overline{AX}$ ; then the length of $\overline{CX}$ is twice this, or . The figure referenced is below:

Chords

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

$AX \cdot BX = CX \cdot DX$

Substituting the appropriate quantities, then solving for :

$t \cdot 20 = 2t \cdot 10$

20 t = 20t

This statement is identically true. Therefore, without further information, we cannot determine the value of - the length of $\overline{AX}$ .

Report an Error

Example Question #1 : Circles

Two chords of a circle, $\overline{AB}$ and $\overline{CD}$ , intersect at a point . $\overline{BX}$ is 12 units longer than $\overline{AX}$ , CX = 8 , and DX = 10 .

Give the length of $\overline{AX}$ (nearest tenth, if applicable)

Possible Answers:

15.0

4.8

3.0

16.8

7.2

Correct answer:

4.8

Explanation:

Let stand for the length of $\overline{AX}$ ; then the length of $\overline{BX}$ is t+12 . The figure referenced is below:

Chords

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

$AX \cdot BX = CX \cdot DX$

Substituting the appropriate quantities, then solving for :

$t \cdot (t+12) = 8 \cdot 10$

$t \cdot t+t \cdot 12 = 8 0$

$t ^{2}+12 t = 8 0$

This quadratic equation can be solved by completing the square; since the coefficient of is 12, the square can be completed by adding

$\left (\frac{12}{2} \right ) ^{2} = 6 ^{2} = 36$

to both sides:

$t ^{2}+12 t + 36 = 8 0+ 36$

Restate the trinomial as the square of a binomial:

$(t+6 )^{2 } = 116$

Take the square root of both sides:

$t+6 = \pm \sqrt{116}$

$t+6 \approx - 10.8$ or $t+6 \approx 10.8$

Either

$t+6 \approx - 10.8$ ,

in which case

$t+6 - 6 \approx - 10.8 - 6$

$t \approx -16.8$ ,

$t+6 \approx 10.8$

in which case

$t+6 - 6 \approx 10.8 - 6$

$t \approx 4.8$ ,

Since is a length, we throw out the negative value; it follows that $t \approx 4.8$ , the correct length of $\overline{AX}$ .

Report an Error

Example Question #1 : Chords

A diameter $\overline{AB}$ of a circle is perpendicular to a chord $\overline{CD}$ at a point .

CD= 20, AX= 8

What is the diameter of the circle?

Possible Answers:

$20\frac{1}{2}$

$33\frac{1}{3}$

Insufficient information is given to answer the question.

Correct answer:

$20\frac{1}{2}$

Explanation:

In a circle, a diameter perpendicular to a chord bisects the chord. This makes the midpoint of $\overline{CD}$ ; consequently, $CX =DX = \frac{1}{2} CD = \frac{1}{2} \cdot 20 = 10$ .

The figure referenced is below:

Chords

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

$AX \cdot BX = CX \cdot DX$

Setting AX= 8, CX =DX = 10 , and solving for :

$8 \cdot BX = 10 \cdot 10$

$8 \cdot BX = 100$

$8 \cdot BX \div 8 = 100 \div 8$

BX = 12.5

AB = AX + BX = 8 + 12.5 = 20.5 ,

the correct length.

Report an Error

Example Question #1 : Circles

Two chords of a circle, $\overline{AB}$ and $\overline{CD}$ , intersect at a point .

CX = AX + 6

BX = 12

DX = 10

Give the length of $\overline{AX}$ .

Possible Answers:

$3 \sqrt{14}$

$2 \sqrt{30}$

Insufficient information is given to answer the question.

Correct answer:

Explanation:

Let t = AX , in which case CX = t+ 6 ; the figure referenced is below (not drawn to scale).

Chords

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

$AX \cdot BX = CX \cdot DX$

Setting AX= t, CX= t+ 6, BX = 12, DX = 10 , and solving for :

$AX \cdot BX = CX \cdot DX$

$t \cdot 12 = (t+6) \cdot 10$

12t = 10t + 60

12t- 10t = 10t + 60 - 10t

2t = 60

$\frac{2t}{2} = \frac{60}{2}$

t = 30 ,

which is the length of $\overline{AX}$ .

Report an Error

Example Question #2 : Circles

A diameter $\overline{AB}$ of a circle is perpendicular to a chord $\overline{CD}$ at point . AX = 12 and BX = 20 . Give the length of $\overline{CD}$ (nearest tenth, if applicable).

Possible Answers:

31.0

16.0

15.5

insufficient information is given to determine the length of $\overline{CD}$ .

32.0

Correct answer:

31.0

Explanation:

A diameter of a circle perpendicular to a chord bisects the chord. Therefore, the point of intersection is the midpoint of $\overline{CD}$ , and

$CX = DX = \frac{1}{2} CD$ .

Let stand for the common length of $\overline{CX}$ and $\overline{DX}$ ,

The figure referenced is below.

Chords

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

$CX \cdot DX = AX \cdot BX$

Set AX = 12 and BX = 20 , and CX = DX = t ; substitute and solve for :

$t \cdot t = 12 \cdot 20$

$t^{2 } = 240$

$t = \sqrt{240} \approx 15.5$

This is the length of $\overline{CX}$ ; the length of $\overline{CD}$ is twice this, so

$CD = 2 \cdot t \approx 2 \cdot 15.5 = 31.0$

Report an Error

Example Question #251 : Geometry

Secant

Figure is not drawn to scale

In the provided diagram, the ratio of the length of $\overarc{BD}$ to that of $\overarc{AC}$ is 7 to 2. Evaluate the measure of $\angle BND$ .

Possible Answers:

$40^{\circ }$

Cannot be determined

$45^{\circ }$

$20^{\circ }$

$25^{\circ }$

Correct answer:

Cannot be determined

Explanation:

The measure of the angle formed by the two secants to the circle from a point outside the circle is equal to half the difference of the two arcs they intercept; that is,

$m \angle BND = \frac{1}{2} ( \overarc{BD} - \overarc{AC})$

The ratio of the degree measure of $\overarc{BD}$ to that of $\overarc{AC}$ is that of their lengths, which is 7 to 2. Therefore,

$\frac{m \overarc{BD} }{m \overarc{AC} } = \frac{7}{2}$

Letting $t = m \overarc{AC}$ :

$\frac{m \overarc{BD} }{t } = \frac{7}{2}$

$\frac{m \overarc{BD} }{t } \cdot t = \frac{7}{2} \cdot t$

$m \overarc{BD} = \frac{7}{2} t$

Therefore, in terms of :

$m \angle BND = \frac{1}{2} \left ( \frac{7}{2} t - t \right ) = \frac{1}{2} \left ( \frac{5}{2} t \right ) =\frac{5}{4} t$

Without further information, however, we cannot determine the value of or that of $m \angle BND$ . Therefore, the given information is insufficient.

Report an Error

Example Question #1 : How To Find The Angle Of Clock Hands

It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?

Possible Answers:

$30^\circ$

$180^\circ$

$90^\circ$

$120^\circ$

$60^\circ$

Correct answer:

$120^\circ$

Explanation:

At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360.

4/12 * 360 = 120 degrees

Report an Error

Example Question #1 : How To Find The Angle Of Clock Hands

If a clock reads 8:15 PM, what angle do the hands make?

Possible Answers:

150^o

240^o

138.5^o

157.5^o

90^o

Correct answer:

157.5^o

Explanation:

A clock is a circle, and a circle always contains 360 degrees. Since there are 60 minutes on a clock, each minute mark is 6 degrees.

$\frac{360^o\ \text{total}}{60\ \text{minutes total}}=6\ \text{degrees per minute}$

The minute hand on the clock will point at 15 minutes, allowing us to calculate it's position on the circle.

$(15\ min)(6)=90^o$

Since there are 12 hours on the clock, each hour mark is 30 degrees.

$\frac{360^o\ \text{total}}{12\ \text{hours total}}=30\ \text{degrees per hour}$

We can calculate where the hour hand will be at 8:00.

$(8\ hr)(30)=240^o$

However, the hour hand will actually be between the 8 and the 9, since we are looking at 8:15 rather than an absolute hour mark. 15 minutes is equal to one-fourth of an hour. Use the same equation to find the additional position of the hour hand.

$240^o+(\frac{1}{4}\ hr)(30)$

240^o+7.5^o

247.5^o

We are looking for the angle between the two hands of the clock. The will be equal to the difference between the two angle measures.

$\angle=247.5^o-90^o=157.5^o$

Report an Error

Example Question #1 : How To Find The Angle Of Clock Hands

What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2?

Possible Answers:

30°

90°

45°

56°

120°

Correct answer:

120°

Explanation:

A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).

Report an Error

Example Question #1 : How To Find The Angle Of Clock Hands

At 5:00 , what angle is between the hour and minute hand on a clock?

Possible Answers:

$160^\circ$

$150^\circ$

$5^\circ$

$120^\circ$

$180^\circ$

Correct answer:

$150^\circ$

Explanation:

At 5:00 , the hour hand is on the and the minute hand is at the . There are spaces on a clock, and these hands are separated by spaces.

Thus, the angle between them is $\frac{5}{12}$ the degrees of the entire clcok, which is $360^\circ$ .

Therefore, we multiply these to get our answer.

$\frac{5}{12}\times \frac{360}{1}$

We can cancel out as we multiply to get:

$\frac{5}{1}\times \frac{30}{1}=150^\circ$

Report an Error

View SAT Mathematics Tutors

Priyanka
Certified Tutor

CUNY Hunter College, Bachelor in Arts, Economics.

View SAT Mathematics Tutors

Karen
Certified Tutor

Bradley University, Bachelor of Science, Multimedia with a minor in Business Administration.

View SAT Mathematics Tutors

Stephen
Certified Tutor

Grinnell College, Bachelor in Arts, Computer Science.

All SAT Math Resources

16 Diagnostic Tests 660 Practice Tests Question of the Day Flashcards Learn by Concept

SAT Math Tutors in Top Cities:

Atlanta SAT Math Tutors, Austin SAT Math Tutors, Boston SAT Math Tutors, Chicago SAT Math Tutors, Dallas Fort Worth SAT Math Tutors, Denver SAT Math Tutors, Houston SAT Math Tutors, Kansas City SAT Math Tutors, Los Angeles SAT Math Tutors, Miami SAT Math Tutors, New York City SAT Math Tutors, Philadelphia SAT Math Tutors, Phoenix SAT Math Tutors, San Diego SAT Math Tutors, San Francisco-Bay Area SAT Math Tutors, Seattle SAT Math Tutors, St. Louis SAT Math Tutors, Tucson SAT Math Tutors, Washington DC SAT Math Tutors

Popular Courses & Classes

MCAT Courses & Classes in Washington DC, GRE Courses & Classes in Boston, GMAT Courses & Classes in Washington DC, ACT Courses & Classes in Chicago, MCAT Courses & Classes in Houston, Spanish Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Atlanta, MCAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Atlanta

Popular Test Prep

MCAT Test Prep in Boston, MCAT Test Prep in Dallas Fort Worth, LSAT Test Prep in San Diego, GRE Test Prep in San Diego, GRE Test Prep in Washington DC, SSAT Test Prep in Chicago, LSAT Test Prep in Los Angeles, GMAT Test Prep in Los Angeles, ISEE Test Prep in Miami, 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)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

SAT Math : Circles

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : Circles

Example Question #1 : Circles

Example Question #1 : Chords

Example Question #1 : Circles

Example Question #2 : Circles

Example Question #251 : Geometry

Example Question #1 : How To Find The Angle Of Clock Hands

Example Question #1 : How To Find The Angle Of Clock Hands

Example Question #1 : How To Find The Angle Of Clock Hands

Example Question #1 : How To Find The Angle Of Clock Hands

All SAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: