Back To Course

Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Kevin Newton*

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

While it sounds very basic, the idea of betweenness has some pretty big implications for math. In this lesson, see not only why it matters, but also how we can use it to help prove basic geometrical problems.

While it sounds unbelievable, one of the central theories of geometry wasn't established until the 20th century. And it is so simple that it only takes a basic understanding of math to understand. Don't you wish you had been the one to notice it? There was a relatively recent development, the idea of **betweenness**, is central to our idea of understanding of math in the real world. In short, it means that if a point B is between points A and C, the length of line AB plus the length of line BC will equal the length of line AC. With it, much of geometry makes sense. Without it, all of geometry falls apart.

Let's back up for a second and make sure that we understand the point of betweenness in the first place. You can draw a straight line between any two points no matter where they are. However, once you add a third point, there is the chance that point is an outlier, and no straight line can connect all three points. By establishing betweenness, we can continue to use many of the advantages of having two points.

Think about a typical polygon, say a parallelogram for example. If we can establish that points on a line segment that forms one of the sides are between each other, then we can say that the lines are straight. Without that knowledge, we suddenly end up with parallelograms that have curves for sides. As you can imagine, this makes formulas, like base x height to find the area, very inaccurate depending on the shape of the curves.

Alongside these fairly obvious benefits of betweeneness there is also the theorem of betweenness that helps us ascertain whether or not the points on a line actually meet the requirements for betweenness. All in all, it's pretty simple, even though it requires some setup.

Let's say you have a line segment with three points, A, B, and C. The line is straight and all three points are on it in the sequence of A, B, and C. The theorem of betweenness tells us that the length of AC is the sum of AB and BC. Now this is only true if B is between A and C, otherwise it would be false. By the same token, if you don't have the length of AB but know BC and AC, you can then find the total length of AB. For thinking that sounds a little circular, don't worry, I've got an example that will help make some sense of it all.

Let's say you have a beautiful view from your bedroom window. Unfortunately, you also have a nosey neighbor whose kitchen window looks right into your room. Rather than cover your window and lose your view, you decide to plant a tree right in between your window and your neighbor's window. You want to plant the tree at the right spot the first time to avoid awkward conversation and betweenness of points will help you do that.

Think of your window as point A and your neighbor's window as point C. We know a straight line can be drawn through any two points, so you treat the line of sight between two windows as line AC. You know your neighbor's window is 50ft away from yours and you want to plant the tree at point B, that is 22ft away from your window. You know if point B falls on the line of sight, you need to also measure its distance from your neighbor's window. You do so and find it's 28 feet away. Now you have two lines; line AB is 22ft and line BC is 28ft. Add those two lengths together and you get 50ft. Since line AC is 50 feet and line AB plus line BC is 50 feet, we know by the theorem of betweenness that point B is between points A and C. Go ahead and plant that tree, you can keep your view and have your privacy too.

In this lesson, we focused on the idea of **betweenness** as a mathematical concept. We defined it as the quality of a point on a line being between two other points on the same line. We saw that while it only has recently been formulated, it is crucial to geometry, as our formulas fall apart without it.

We also learned that the **theorem of betweenness** lets us explain this concept using math, stating that if a line has points A, B, and C, that the length of AC is the sum of AB and BC.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
3 in chapter 1 of the course:

Back To Course

Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

- Undefined Terms of Geometry: Concepts & Significance 5:23
- Line Segments & Rays: Definition & Measurement 3:59
- Betweenness of Points: Definition & Problems 4:08
- How to Use The Distance Formula 5:27
- Points, Lines & Angles in Geometry 5:13
- What is an Obtuse Angle? - Definition & Examples 2:35
- What is an Acute Angle? - Definition, Description & Examples 1:45
- What is a Right Angle? - Definition & Formula 3:19
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Constructing an Angle Bisector in Geometry 3:36
- Vertical Angles & Complementary Angles: Definition & Examples 4:17
- Congruent Angles: Definition & Examples 3:52
- What Are Adjacent Angles? - Definition & Examples 3:21
- Parallel, Perpendicular and Transverse Lines 6:06
- What Are Polygons? - Definition and Examples 4:25
- Regular Polygons: Definition & Parts 6:01
- Finding the Perimeter of Polygons 5:19
- Perimeter of Triangles and Rectangles 8:54
- Go to Glencoe Geometry Chapter 1: Points, Lines, Planes, and Angles

- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- U.S. Politics & Civics Lesson Plans
- US History - Civil War: Lesson Plans & Resources
- HESI Admission Assessment Exam: Factors & Multiples
- HESI Admission Assessment Exam: Probability, Ratios & Proportions
- HESI Admission Assessment Exam: 3D Shapes
- HESI Admission Assessment Exam: Punctuation
- HESI Admission Assessment Exam: Linear Equations, Inequalities & Functions
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison
- TACHS Prep Product Comparison
- Top 50 Blended Learning High Schools
- EPPP Prep Product Comparison

- History of Sparta
- Realistic vs Optimistic Thinking
- How Language Reflects Culture & Affects Meaning
- Logical Thinking & Reasoning Questions: Lesson for Kids
- Exceptions to the Octet Rule in Chemistry
- Database Hacking: Attack Types & Defenses
- Pride and Prejudice Discussion Questions
- Quiz & Worksheet - Frontalis Muscle
- Quiz & Worksheet - Dolphin Mating & Reproduction
- Octopus Diet: Quiz & Worksheet for Kids
- Quiz & Worksheet - Fezziwig in A Christmas Carol
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Narrative Essay Topics for Teachers
- Grammar Worksheets & Practice

- College Calculus Textbook
- OSAT Chemistry (CEOE) (004): Practice & Study Guide
- OSAT Early Childhood Education (CEOE) (105): Practice & Study Guide
- Common Core ELA Grade 7 - Writing: Standards
- Blended Learning Models and Solutions for Teachers
- Cambridge Pre-U Mathematics: Trigonometric Graphs
- Virginia SOL - World History & Geography 1500-Present: The Scientific Revolution
- Quiz & Worksheet - Improving Health Education Program Performance
- Quiz & Worksheet - Foraging & Pastoral Nomadic Societies
- Quiz & Worksheet - Veterinarians in Society
- Quiz & Worksheet - Julius Caesar's Wife
- Quiz & Worksheet - African Religious Systems

- Cars & Culture in the U.S. in the 1950s
- Swimming Skills & Biomechanics
- Student Loan Forgiveness for Teachers in Texas
- Types of Government Lesson Plan
- Virginia Homeschool Laws
- STEM & Next Generation Science Standards
- Kentucky Science Standards for 4th Grade
- What is the PSAT 8/9? - Information, Structure & Scoring
- Paraprofessional Certification in Texas
- Virginia Homeschool Laws
- GRE to GMAT Conversion
- Georgia Physical Science Standards

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject