Articles containing proofs | Metric geometry | Geometric inequalities | Linear algebra | Theorems in geometry
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that with equality only in the degenerate case of a triangle with zero area.In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms): where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute values. In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints. The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces. (Wikipedia).
This video states and investigates the triangle inequality theorem. Complete Video List: http://www.mathispower4u.yolasite.com
From playlist Relationships with Triangles
This video focuses on how to use the Triangle Inequality Theorem to determine if three lengths can make up the sides of a triangle. More specifically, I show students how to quickly use the Triangle Inequality Theorem to determine if three values can make up the sides of a triangle. Your
From playlist Geometry
I introduce Inequalities in 1 Triangle in which we look at how the lengths of the sides are related to the size of the angles that are opposite those sides. EXAMPLES AT 2:34 6:04 9:55 11:43 14:25 15:30 At minute 2:07 I misspoke, 20+30 is 50 of course:D Find free review test, useful notes
From playlist Geometry
Triangle Inequality Theorem Possible Values of x
Learn how to use the triangle inequality theorem to find possible values of x in this video math tutorial by Mario's Math Tutoring. We discuss what is the triangle inequality theorem as well as solve an algebraic problem given 3 sides of a triangle. 0:10 What is the Triangle Inequality
From playlist Geometry
Triangle Inequality Theorem - Example | Don't Memorise
To learn more about Triangles enroll in our full course now: https://infinitylearn.com/microcourses?utm_source=youtube&utm_medium=Soical&utm_campaign=DM&utm_content=BiagrTl2y4o&utm_term=%7Bkeyword%7D In this video, we will learn: 0:00 Introduction 0:29 triangle inequality theorem 0:46 co
From playlist Middle School Math - Triangles
What is the Pythagorean Inequality Theorem
👉 Learn all about classifying triangles. A triangle is a closed figure with three sides. A triangle can be classified based on the length of the sides or based on the measure of the angles. To classify a triangle based on the length of the sides, we have: equilateral (3 sides are equal), i
From playlist Triangles
Proof: Triangle Inequality Theorem | Real Analysis
The absolute value of a sum is less than or equal to the sum of the absolute values for any two real numbers. That is: |a+b| is less than or equal to |a|+|b|. This is called the triangle inequality. It's very useful in real analysis and we'll prove it in today's lesson! The name of the the
From playlist Real Analysis
Triangle Inequality In this video, I define the concept of an absolute value and use it to prove the triangle inequality in R, which is the most important inequality that we'll use in analysis. Enjoy! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmC
From playlist Real Numbers
17. Graph limits IV: inequalities between subgraph densities
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Among all graphs with a given edge density, which graph h
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
Triangle Inequality Solve for x
Learn how to use the triangle inequality theorem to solve for x in this free math video tutorial example by Mario's Math Tutoring. 0:08 Example 1 Find the Value of x in the Triangle with sides 2x+1, 3x-4, x+7 0:15 What is the Triangle Inequality Theorem 0:51 Writing 3 Inequalities to Solv
From playlist Geometry
Max/Min Value of |z| (2 of 2: Triangle inequality)
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From playlist Using Complex Numbers
Proof: Reverse Triangle Inequality Theorem | Real Analysis
The reverse triangle inequality tells us how the absolute value of the difference of two real numbers relates to the absolute value of the difference of their absolute values. In particular, |x-y| is greater than or equal to | |x| - |y| |. We'll prove this result in today's lesson. We'll n
From playlist Real Analysis
Complex Analysis Episode 5: Triangle Inequality and Upper Bounds
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From playlist Complex Analysis
Inequality Proof using Both the Triangle Inequality and Reverse Triangle Inequality
Inequality Proof using Both the Triangle Inequality and Reverse Triangle Inequality
From playlist Advanced Calculus
An AMC geometry problem with the triangle inequality.
From playlist Challenge Problems
Proofs from calculus -- Proof Writing 20
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From playlist Proof Writing
Complex Analysis 01: Inequalities
The two basic inequalities, and an application
From playlist MATH2069 Complex Analysis
Norms in inner product spaces. Othogonality. The Cauchy-Schwarz Inequality. The Triangle Inequality. The Parallelogram Equality.
From playlist Linear Algebra Done Right
Coloring with Primes–Monsky's Theorem #some2
Thank you for watching! We hope you enjoyed this video. This video was created as our entry to some2. We had a good time making this video and learned a lot of cool math in the process. Below are some of the resources we used and recommend if you found this interesting: More math related
From playlist Summer of Math Exposition 2 videos
What is the Triangle Inequality Theorem - Congruent Triangles
👉 Learn about congruent triangles theorems. Two or more triangles are said to be congruent if they have the same shape and size. There are many methods to determine whether two triangles are congruent. Some of the methods include: (1) The HL (Hypothenuse Leg) theorem:- The hypothenuse le
From playlist Congruent Triangles