Call Now to Set Up Tutoring:

(888) 888-0446

SAT Math : Coordinate Geometry

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 #11 : How To Find The Length Of A Line With Distance Formula

In terms of , give the length of a segment on the coordinate plane with endpoints $\left (- \frac{1}{4} , N \right )$ and $\left (N, - \frac{1}{4} \right )$ .

Possible Answers:

$\sqrt{2N^{2}- N}$

$\sqrt{2N^{2}+N+\frac{1}{8}}$

$\sqrt{2N^{2}-N+\frac{1}{8}}$

$\sqrt{2N^{2}+ \frac{1}{8}}$

$\sqrt{2N^{2}+ N}$

Correct answer:

$\sqrt{2N^{2}+N+\frac{1}{8}}$

Explanation:

The length of a segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ can be calculated using the distance formula:

$d = \sqrt{(x_{2}- x_{1})^{2} + (y_{2}- y_{1})^{2} }$

Setting $x_{1} = y_{2} = -\frac{1}{4}$ and $x_{2} = y_{1} = N$ , and substituting:

$d = \sqrt{ \left (N - \left (- \frac{1}{4} \right )\right ) ^{2} + \left ( - \frac{1}{4} - N \right ) ^{2}}$

$d = \sqrt{ \left (N + \frac{1}{4} \right ) ^{2} + \left ( - N - \frac{1}{4} \right ) ^{2}}$

The binomials can be rewritten using the perfect square trinomial pattern:

$d = \sqrt{ \left [N^{2}+ 2 \cdot N \cdot \frac{1}{4} + \left ( \frac{1}{4}\right ) ^{2} \right ] + \left [(-N)^{2} - 2 \cdot (-N) \cdot\left ( - \frac{1}{4}\right ) + \left ( - \frac{1}{4}\right ) ^{2} \right ] }$

$d = \sqrt{ \left ( N^{2}+ \frac{1}{2} N + \frac{1}{16} \right ) +\left ( N^{2}+ \frac{1}{2} N + \frac{1}{16} \right ) }$

Simplify and collect like terms:

$d = \sqrt{ N^{2}+ N^{2}+ \frac{1}{2} N + \frac{1}{2} N + \frac{1}{16} + \frac{1}{16} }$

$= \sqrt{2 N^{2}+ N + \frac{1}{8} }$

Report an Error

Example Question #11 : How To Find The Length Of A Line With Distance Formula

Give the length, in terms of , of a line segment on the coordinate plane whose endpoints are (N, N) and $\left ( \frac{1}{3}, -\frac{1}{3} \right )$ .

Possible Answers:

$\sqrt{2N^{2}+\frac{2}{9}}$

$\sqrt{2N^{2}+\frac{4}{3}N+\frac{2}{9}}$

$\sqrt{2N^{2}-\frac{4}{3}N+\frac{2}{9}}$

$\sqrt{2N^{2}-\frac{4}{3}N }$

$\sqrt{2N^{2}+\frac{4}{3}N }$

Correct answer:

$\sqrt{2N^{2}+\frac{2}{9}}$

Explanation:

The length of a segment with endpoints $(x_{1},y_{1}) , (x_{2},y_{2})$ can be calculated using the distance formula

$d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }$

Substituting $x_{1} = \frac{1}{3} ,y_{1} =-\frac{1}{3}, x_{2} = y_{2}= N$ , and simplifying"

$d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2} }$

$= \sqrt{\left (N - \frac{1}{3}\right )^{2}+ \left (N - \left ( - \frac{1}{3} \right ) \right )^{2} }$

$= \sqrt{\left (N - \frac{1}{3}\right )^{2}+ \left (N + \frac{1}{3} \right ) ^{2} }$

$= \sqrt{ \left [ N^{2} - 2 \cdot N \cdot \frac{1}{3} + \left ( \frac{1}{3}\right )^{2} \right ]+ \left [ N ^{2} + 2 \cdot N \cdot \frac{1}{3} + \left ( \frac{1}{3}\right )^{2} \right ] }$

$= \sqrt{ \left ( N^{2} - \frac{2}{3}N + \frac{1}{9} \right ) + \left ( N^{2} + \frac{2}{3}N + \frac{1}{9} \right ) }$

$= \sqrt{ N^{2} + N^{2} - \frac{2}{3}N + \frac{2}{3}N + \frac{1}{9} + \frac{1}{9} }$

$= \sqrt{ 2N^{2} +\frac{2} {9} }$

Report an Error

Example Question #21 : Distance Formula

Find the length of a line segment whose endpoints are located at the coordinates (7,-4) and (-5,1) .

Possible Answers:

$2\sqrt{42}$

$\sqrt{13}$

$4\sqrt{10}$

Correct answer:

Explanation:

According to the distance formula, the linear distance between two points (x_a, y_a) and (x_b,y_b) is given by

$d=\sqrt{(x_a-x_b)^2+(y_a-y_b)^2}$ ,

which is a generalization of the Pythagorean Theorem c^2=a^2+b^2 for any two points on the coordinate plane.

The two points given in this problem are (7,-4) and (-5,1) . Although the question asks for the length of a line segment given its two endpoints, the distance formula is applicable here because the geometric meaning of "distance between two points" is the length of a line segment. Hence, we can use the distance formula to determine the distance between these two endpoints as shown:

$d=\sqrt{((7)-(-5))^2+((-4)-(1))^2}$

$\Rightarrow d=\sqrt{(12)^2+(-5)^2}$

$\Rightarrow d=\sqrt{144+25}$

$\Rightarrow d=\sqrt{169}$

$\Rightarrow d=13$

Hence, the line segment bounded by the points (7,-4) and (-5,1) is units long.

Report an Error

Example Question #21 : How To Find The Length Of A Line With Distance Formula

One line has four collinear points in order from left to right A, B, C, D. If AB = 10’, CD was twice as long as AB, and AC = 25’, how long is AD?

Possible Answers:

50'

45'

35'

30'

40'

Correct answer:

45'

Explanation:

AB = 10 ’

BC = AC – AB = 25’ – 10’ = 15’

CD = 2 * AB = 2 * 10’ = 20 ’

AD = AB + BC + CD = 10’ + 15’ + 20’ = 45’

Report an Error

Example Question #1 : Midpoint Formula

The midpoint of line segment AB is (2, -5). If the coordinates of point A are (4, 4), what are the coordinates of B?

Possible Answers:

(0, -13)

(6, 14)

(3, -0.5)

(6, 13)

(0, -14)

Correct answer:

(0, -14)

Explanation:

The fastest way to find the missing endpoint is to determine the distance from the known endpoint to the midpoint and then performing the same transformation on the midpoint. In this case, the x-coordinate moves from 4 to 2, or down by 2, so the new x-coordinate must be 2-2 = 0. The y-coordinate moves from 4 to -5, or down by 9, so the new y-coordinate must be -5-9 = -14.

An alternate solution would be to substitute (4,4) for (x₁,y₁) and (2,-5) for (x,y) into the midpoint formula:

x=(x₁+x₂)/2

y=(y₁+y₂)/2

Solving each equation for (x₂,y₂) yields the solution (0,-14).

Report an Error

Example Question #2 : Midpoint Formula

Point A is (5, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

Possible Answers:

(12, –11)

(8.5, –2)

None of the other answers

(22, –9)

(29, –15)

Correct answer:

(29, –15)

Explanation:

Point A is (5, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

We need to use our generalized midpoint formula:

MP = ( (5 + x)/2, (7 + y)/2 )

Solve each separately:

(5 + x)/2 = 17 → 5 + x = 34 → x = 29

(7 + y)/2 = –4 → 7 + y = –8 → y = –15

Therefore, B is (29, –15).

Report an Error

Example Question #1 : Midpoint Formula

Line segment AB has an endpoint, A, located at (10,-1) , and a midpoint at (10,0) . What are the coordinates for point B of segment AB?

Possible Answers:

(5,1)

(5,-0.5)

(10,-0.5)

The second endpoint cannot exist

(10,1)

Correct answer:

(10,1)

Explanation:

With an endpoint A located at (10,-1), and a midpoint at (10,0), we want to add the length from A to the midpoint onto the other side of the segment to find point B. The total length of the segment must be twice the distance from A to the midpoint.

A is located exactly one unit below the midpoint along the y-axis, for a total displacement of (0,1). To find point B, we add (10+0, 0+1), and get the coordinates for B: (10,1).

Report an Error

Example Question #3 : Midpoint Formula

Solve each problem and decide which is the best of the choices given.

What is the distance between the points (1,6) and (-7,3) on a standard coordinate plane?

Possible Answers:

$\sqrt{39}$

$\sqrt{73}$

$\sqrt{55}$

Correct answer:

$\sqrt{73}$

Explanation:

Make a triangle. The points are 8 units apart on the -axis, and units apart on the -axis. Then use the Pythagorean Theorem to find the distance of the hypotenuse, which ends up being $\sqrt{73}$ .

Another way to solve this problem is to use the distance formula,

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Plugging in the two points we get,

$d=\sqrt{(1--7)^2+(6-3)^2}$

$d=\sqrt{(8)^2+(3)^2}$

$d=\sqrt{64+9}$

$d=\sqrt{73}$

Report an Error

Example Question #1 : How To Find The Midpoint Of A Line Segment

A line segment has endpoints (0,4) and (5,6). What are the coordinates of the midpoint?

Possible Answers:

(0,6)

(0,4)

(2.5,5)

(2.5,-5)

(3,9)

Correct answer:

(2.5,5)

Explanation:

A line segment has endpoints (0,4) and (5,6). To find the midpoint, use the midpoint formula:

X: (x₁+x₂)/2 = (0+5)/2 = 2.5

Y: (y₁+y₂)/2 = (4+6)/2 = 5

The coordinates of the midpoint are (2.5,5).

Report an Error

Example Question #1 : Midpoint Formula

Find the midpoint between (-3,7) and (5,-9)

Possible Answers:

(4,-1)

(1,-8)

(-1,-1)

(1,-1)

(4,-8)

Correct answer:

(1,-1)

Explanation:

You can find the midpoint of each coordinate by averaging them. In other words, add the two x coordinates together and divide by 2 and add the two y coordinates together and divide by 2.

x-midpoint = (-3 + 5)/2 = 2/2 = 1

y-midpoint = (7 + -9)/2 = -2/2 = -1

(1,-1)

Report an Error

View SAT Mathematics Tutors

Troy
Certified Tutor

Rice University, Bachelor in Arts, Kinesiology.

View SAT Mathematics Tutors

Irina
Certified Tutor

New York University, Bachelors, Biochemistry.

View SAT Mathematics Tutors

Kenneth
Certified Tutor

North Georgia University, Bachelors, Accounting. Brenau University, Masters, Business.

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

SAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Atlanta, SAT Courses & Classes in Philadelphia, SAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Los Angeles, Spanish Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Chicago, GMAT Courses & Classes in Miami, GRE Courses & Classes in Atlanta, SSAT Courses & Classes in Atlanta

Popular Test Prep

LSAT Test Prep in Seattle, SSAT Test Prep in New York City, ISEE Test Prep in Miami, ACT Test Prep in Atlanta, MCAT Test Prep in Phoenix, ACT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Chicago, LSAT Test Prep in Washington DC, MCAT Test Prep in Los Angeles, SSAT Test Prep in San Diego

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 : Coordinate Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #11 : How To Find The Length Of A Line With Distance Formula

Example Question #11 : How To Find The Length Of A Line With Distance Formula

Example Question #21 : Distance Formula

Example Question #21 : How To Find The Length Of A Line With Distance Formula

Example Question #1 : Midpoint Formula

Example Question #2 : Midpoint Formula

Example Question #1 : Midpoint Formula

Example Question #3 : Midpoint Formula

Example Question #1 : How To Find The Midpoint Of A Line Segment

Example Question #1 : Midpoint Formula

All SAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: