How To Find The Perpendicular Bisector Of Two Points - WikiHow

Skip to ContentQuizzes

• PRO Courses Hot Guides Tech Help Pro Expert Videos About wikiHow Pro Upgrade
• QUIZZES All Quizzes Hot Love Quizzes Personality Quizzes Trivia Quizzes Taylor Swift Quizzes
• EDIT Edit this Article
• EXPLORE Tech Help ProAbout UsRandom ArticleQuizzes Request a New ArticleCommunity DashboardTrendingForums Arts and EntertainmentArtworkBooksMoviesComputers and ElectronicsComputersPhone SkillsTechnology HacksHealthMen's HealthMental HealthWomen's HealthRelationshipsDatingLoveRelationship Issues Hobbies and CraftsCraftsDrawingGamesEducation & CommunicationCommunication SkillsPersonal DevelopmentStudyingPersonal Care and StyleFashionHair CarePersonal Hygiene Quizzes Love Quizzes Personality Quizzes Fun Games ForumsArts and EntertainmentFinance and BusinessHome and Garden Relationship QuizzesCars & Other VehiclesFood and EntertainingPersonal Care and Style Sports and FitnessComputers and ElectronicsHealthPets and Animals TravelEducation & CommunicationHobbies and CraftsPhilosophy and Religion Work WorldFamily LifeHolidays and TraditionsRelationships Youth
• LOG IN

  Log in

  Social login does not work in incognito and private browsers. Please log in with your username or email to continue. Facebook Google wikiHow Account No account yet? Create an account
• RANDOM

• Home
• Random
• Browse Articles
• TrendingNew
• Quizzes & Games
• All QuizzesHot
• Love Quizzes
• Personality Quizzes
• Fun Games
• Dating Simulator
• Learn Something New
• Forums
• Courses
• Happiness Hub
• Explore More
• Support wikiHow
• About wikiHow

• Log in / Sign up

Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow

• Categories
• Education and Communications
• Studying
• Mathematics
• Geometry
• Coordinate Geometry

How to Find the Perpendicular Bisector of Two Points Download Article Reviewed by Grace Imson, MA

Last Updated: March 10, 2025 Fact Checked

Download Article

• Gathering Information
|
• Calculating the Equation of the Line
|
• Video
|
• Q&A

|Show more |Show less X

This article was reviewed by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,127,378 times.

A perpendicular bisector is a line that cuts a line segment connecting two points exactly in half at a 90 degree angle. To find the perpendicular bisector of two points, all you need to do is find their midpoint and negative reciprocal, and plug these answers into the equation for a line in slope-intercept form. If you want to know how to find the perpendicular bisector of two points, just follow these steps.

Steps

Method 1 Method 1 of 2:

Gathering Information

Download Article

1. 1 Find the midpoint of the two points. To find the midpoint of two points, simply plug them into the midpoint formula: [(x1 + x2)/2,( y1 + y2)/2]. This means that you're just finding the average of the x and y coordinates of the two sets of points, which leads you to the midpoint of the two coordinates. Let's say we're working with the (x1, y1) coordinates of (2, 5) and the (x2, y2) coordinates of (8, 3). Here's how you find the midpoint for those two points:[1]
   • [(2+8)/2, (5 +3)/2] =
   • (10/2, 8/2) =
   • (5, 4)
   • The coordinates of the midpoint of (2, 5) and (8, 3) are (5, 4).
2. 2 Find the slope of the two points. To find the slope of the two points, simply plug the points into the slope formula: (y2 - y1) / (x2 - x1). The slope of a line measures the distance of its vertical change over the distance of its horizontal change. Here's how to find the slope of the line that goes through the points (2, 5) and (8, 3):[2]
   • (3-5)/(8-2) =
   • -2/6 =
   • -1/3
     • The slope of the line is -1/3. To find this slope, you have to reduce 2/6 to its lowest terms, 1/3, since both 2 and 6 are evenly divisible by 2.
   Advertisement
3. 3 Find the negative reciprocal of the slope of the two points. To find the negative reciprocal of a slope, simply take the reciprocal of the slope and change the sign. You can take the negative reciprocal of a number simply by flipping the x and y coordinates and changing the sign. The normal reciprocal of 1/2 would be 2/1 and the negative reciprocal is -2/1, or just -2; the reciprocal of -4 is 1/4.[3]
   • The negative reciprocal of -1/3 is 3 because 3/1 is the reciprocal of 1/3 and the sign has been changed from negative to positive.
Advertisement

Method 2 Method 2 of 2:

Calculating the Equation of the Line

Download Article

1. 1 Write the equation of a line in slope-intercept form. The equation of a line in slope-intercept form is y = mx + b where any x and y coordinates in the line are represented by the "x" and "y," the "m" represents the slope of the line, and the "b" represents the y-intercept of the line. The y-intercept is where the line intersects the y-axis. Once you write down this equation, you can begin to find the equation of the perpendicular bisector of the two points.[4]
2. 2 Plug the negative reciprocal of the original slope into the equation. The negative reciprocal of the slope of the points (2, 5) and (8, 3) was 3. The "m" in the equation represents the slope, so plug the 3 into the "m" in the equation of y = mx + b.[5]
   • 3 --> y = mx + b =
   • y = 3x + b
3. 3 Plug the points of the midpoint into the line. You already know that the midpoint of the points (2, 5) and (8, 3) is (5, 4). Since the perpendicular bisector runs through the midpoint of the two lines, you can plug the coordinates of the midpoint into the equation of the line. Simply plug in (5, 4) into the x and y coordinates of the line.[6]
   • (5, 4) ---> y = 3x + b =
   • 4 = 3(5) + b =
   • 4 = 15 + b
4. 4 Solve for the intercept. You have found three of the four variables in the equation of the line. Now you have enough information to solve for the remaining variable, "b," which is the y-intercept of this line. Simply isolate the variable "b" to find its value. Just subtract 15 from both sides of the equation.[7]
   • 4 = 15 + b =
   • -11 = b
   • b = -11
5. 5 Write the equation of the perpendicular bisector. To write the equation of the perpendicular bisector, you simply have to plug in the slope of the line (3) and the y-intercept (-11) into the equation of a line in slope-intercept form. You should not plug in any terms into the x and y coordinates, because this equation will allow you to find any coordinate on the line by plugging in either any x or any y coordinate.[8]
   • y = mx + b
   • y = 3x - 11
   • The equation for the perpendicular bisector of the points (2, 5) and (8, 3) is y = 3x - 11.
Advertisement

Calculator, Practice Problems, and Answers

Sample Find the Perpendicular Bisector of Two Points Calculator Sample Find the Perpendicular Bisector of Two Points Practice Problems Sample Find the Perpendicular Bisector of Two Points Practice Answers

Community Q&A

Search Add New Question

• Question What if one of the numbers is a negative, and when I try to find the midpoint it gives me 0? The points are A(-4,4) and B(4,8). Donagan Top Answerer Zero is correct. The x-value midway between -4 and +4 is 0. The y-value midway between 8 and 4 is 6. So the midpoint is (0,6). Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 14 Helpful 55
• Question What is the exact length of a line joining the points (-12,3) and (8,4)? Donagan Top Answerer Visualize a right triangle, the hypotenuse of which is the line joining the two points. One leg of the triangle has this length: 8 - (-12) = 8 + 12 = 20. The length of the other leg is 4 - 3 = 1. The hypotenuse has this length: √(20² + 1²) = √401. Thus, the distance between the points is 20.025. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 13 Helpful 30
• Question Is a bisector the same as the negative reciprocal of the line? Donagan Top Answerer No. A line's negative reciprocal would be another line perpendicular to it but not necessarily bisecting it. Thanks! We're glad this was helpful. Thank you for your feedback. If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow Yes No Not Helpful 12 Helpful 22

See more answers Ask a Question 200 characters left Include your email address to get a message when this question is answered. Submit Advertisement

Video

Tips

Submit a Tip All tip submissions are carefully reviewed before being published Name Please provide your name and last initial Submit Thanks for submitting a tip for review!

You Might Also Like

How toFind the Equation of a Perpendicular Line How toFind the Equation of a Line How to Calculate Slope, Y-Intercept, and X-Intercept How toFind the Equation of a Perpendicular Line Given an Equation and Point How toFigure out if Two Lines Are Parallel How toFind the Slope of a Line Using Two Points How toFind the Midpoint of a Line Segment 3 Methods to Find and Solve Circumcenter How to Calculate the Slope of a Line How toFind the Y Intercept The Ultimate Guide to Finding the Slope of a Line Using Linear Equations to Find Ordered Pairs (with Practice Problems) How toFind the X Intercept How toConstruct a Perpendicular Line to a Given Line Through Point on the Line Advertisement

References

1. ↑ https://www.calculatorsoup.com/calculators/geometry-plane/midpoint-calculator.php
2. ↑ https://www.mathwarehouse.com/algebra/linear_equation/slope-of-a-line.php
3. ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut28_parpen.htm
4. ↑ https://www.mathplanet.com/education/algebra-1/visualizing-linear-functions/the-slope-intercept-form-of-a-linear-equation
5. ↑ https://www.mathplanet.com/education/algebra-1/visualizing-linear-functions/the-slope-intercept-form-of-a-linear-equation
6. ↑ https://www.slopecalculator.io/perpendicular-bisector-calculator
7. ↑ https://www.slopecalculator.io/perpendicular-bisector-calculator
8. ↑ https://mathspace.co/textbooks/syllabuses/Syllabus-1059/topics/Topic-20590/subtopics/Subtopic-268708/

About This Article

Reviewed by: Grace Imson, MA Math Teacher This article was reviewed by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 1,127,378 times. 20 votes - 88% Co-authors: 31 Updated: March 10, 2025 Views: 1,127,378 Categories: Coordinate Geometry Article SummaryX

To find the perpendicular bisector of 2 points, find the midpoint of the 2 points by using the midpoint formula. Then, find the slope of the 2 points by using the slope formula, and find the negative reciprocal of the slope by taking the reciprocal and changing the sign. Write the equation of the line in point-slope form using the negative reciprocal and the midpoint. Solve the equation for the intercept to find the perpendicular bisector. For more information, including the formulas for finding midpoint and slope, scroll down! Did this summary help you?YesNo

In other languages Italian Spanish German Russian French Dutch Arabic Chinese Thai Japanese Hindi Turkish Korean

• Print
• Send fan mail to authors

Thanks to all authors for creating a page that has been read 1,127,378 times.

Reader Success Stories

• 

  Anonymous

  Oct 23, 2016

  "Great explanation! My teacher didn't explain it quite like this. Thank you so much, I appreciated this..." more

More reader stories Hide reader stories Share your story

Did this article help you?

Yes No Advertisement Cookies make wikiHow better. By continuing to use our site, you agree to our cookie policy. Reviewed by: Grace Imson, MA Math Teacher Co-authors: 31 Updated: March 10, 2025 Views: 1,127,378 88% of readers found this article helpful. 20 votes - 88% Click a star to add your vote

Anonymous

Oct 23, 2016

"Great explanation! My teacher didn't explain it quite like this. Thank you so much, I appreciated this..." more

Anonymous

Aug 20, 2017

"Thanks a lot, wikiHow! Now I know where to come first if I'm looking to do something I don't know how to..." more

Khadija Kolsawala

Jun 5, 2017

"This helped me in solving my homework questions. Saved a lot time in searching here and there. Thank you."

Anna Kh

May 3, 2018

"Everything helped, basically. It was really easy to understand and memorize. Thank you! :) "

Anonymous

Nov 12, 2017

"It was very clear and simple, with illustrations of the equations written out. I love it!" Share yours! More success stories Hide success stories

Quizzes & Games

What Age Is My Brain QuizTake QuizCognitive TestTake QuizIQ Test For KidsTake QuizIQ TestTake QuizMusic Notes & Symbols TestTake QuizHow Intelligent Am I QuizTake Quiz

You Might Also Like

How toFind the Equation of a Perpendicular LineHow toFind the Equation of a LineHow to Calculate Slope, Y-Intercept, and X-InterceptHow toFind the Equation of a Perpendicular Line Given an Equation and Point

Featured Articles

What Does It Mean If Someone Tries to Kill You in Your Dream?200+ Best Questions to Ask Someone You Like (Texting & IRL)How Beautiful Am I Quiz13 Types of Hairlines for Men and Women (And How to Maintain Yours)Which of the 8 Face Shapes Do You Have? Helpful Tips & Style GuidanceMental Illness Test

Trending Articles

Let Us Rate You 1-10: Attractiveness Scale QuizCan We Guess If You're Single or Taken?How toMake OutAm I a Gigachad or Totally Chopped?How Rare Is My Birthday QuizHow Nonchalant Am I Quiz

Featured Articles

How toEnjoy the HolidaysHow to Get 250 Grams of Protein Daily: Best Foods, Meal Plans & MoreAm I Girly Quiz200 Fun, Flirty, & Deep Questions to Get to Know a Guy BetterHow to Tighten Skin Under Your Chin: Natural or Surgical OptionsWhat Would My Nickname Be Quiz

Featured Articles

21 Common Hand Signs & Gestures (& What They Mean)170+ Best Useless Facts That Sound Like LiesIs He Interested In Me Quiz15 Signs Your Ex Misses You (and What to Do About It)How to Observe Hanukkah (From Candle Lighting to Latke Frying)165+ "You Are Special" Quotes For Friends, Family, Partners & More

Watch Articles

How toWrap a Round GiftHow toSteam in an OvenHow to Make Plant-Based Meat: 4 Incredible RecipesHow toClean a Silver Necklace4 Easy Ways to Open a Can of SodaHow toPart Your Hair

Trending Articles

180 Good Comebacks & Savage Roasts to Win Any ArgumentAm I a Furry QuizHave I Met My Soulmate Yet QuizRandom Fortune GeneratorWhat Should My Name Really Be QuizWhat Are the Gay Male Body Types (And Which Do You Have?)

Quizzes & Games

Am I a Good Friend QuizTake QuizAm I a Genius QuizTake QuizAm I Smarter than a 5th Grader QuizTake QuizDo I Have Common Sense QuizTake QuizAm I Good at Flirting QuizTake QuizAm I Misogynistic? Get Answers HereTake Quiz

• Categories
• Education and Communications
• Studying
• Mathematics
• Geometry
• Coordinate Geometry

wikiHow Newsletter You're all set! Helpful how-tos delivered toyour inbox every week! Sign me up! By signing up you are agreeing to receive emails according to our privacy policy.

• Home
• About wikiHow
• Experts
• Jobs
• Contact Us
• Site Map
• Terms of Use
• Privacy Policy
• Do Not Sell or Share My Info
• Not Selling Info
• Contribute

Follow Us

×

Get all the best how-tos!

Sign up for wikiHow's weekly email newsletter

Subscribe You're all set! X --492