The triangle inequality theorem is one of the important mathematical principles that is used across various branches of mathematics. In real life, civil engineers use the triangle inequality theorem since their area of work deals with surveying, transportation, and urban planning. The triangle inequality theorem helps them to calculate the unknown lengths and have a rough estimate of various dimensions. In this article, let's learn about the triangle inequality theorem and its proof using solved examples. Show
What is Triangle Inequality?The Triangle Inequality (theorem) says that in any triangle, the sum of any two sides must be greater than the third side. For example, consider the following ∆ABC:  According to the Triangle Inequality theorem:
How Does Triangle Inequality Work?An easy way to understand how the triangle inequality theorem works in any ∆ABC is to imagine yourself walking along the sides of the triangle. If you have to go from A to B, for example, the shortest path will be segment AB. If you first go to C and then to B, the distance you cover, AC + CB, will surely be greater than AB. Alternatively, let's try and understand the Triangle Inequality theorem through construction. Suppose that you are given three lengths: x, y, and z. You are asked to construct a triangle with these sides. You proceed as follows:
Observe carefully that the two arcs will intersect only if the sum of the radii of the two arcs is greater than the distance between the centers of the arc. In other words, to be able to draw a triangle: x + y must be greater than z This means, for example, there can be no triangle with sides 2 units, 2 units, and 5 units, because: 2 + 2 < 5 This is how triangle inequality works. Triangle Inequality ProofLet us now discuss the Triangle Inequality proof. Consider the following triangle, ∆ABC: We need to prove that AB + AC > BC. Proof: Extend BA to point D such that AD = AC, and join C to D, as shown below: We note that ∠ACD = ∠D, which means that in ∆ BCD, ∠BCD > ∠D. Sides opposite larger angles are larger, and thus: BD > BC AB + AD > BC AB + AC > BC (because AD = AC) This completes our proof. We can additionally conclude that in a triangle:
Related Topics:Check out these interesting articles to learn more about triangle inequality and its related topics. Important NotesHere is a list of a few points that should be remembered while studying triangle inequality:
go to slidego to slide
Have questions on basic mathematical concepts? Become a problem-solving champ using logic, not rules. Learn the why behind math with our Cuemath’s certified experts. Book a Free Trial Class
FAQs on Triangle InequalityAs per the triangle inequality theorem, the sum of the lengths of any two sides of a triangle is greater than the length of the third side. What are the Applications of Triangle Inequality?The triangle inequality theorem is one of the most important mathematical principles that is used across various branches of mathematics. It is a useful tool for checking if a given set of three dimensions will form a triangle or not. In real life, mapping applications like Google Maps make use of triangle inequalities to calculate unknown distances between places. How Can Three Equal Sides Form a Triangle as per Triangle Inequality?When three equal sides form a triangle, they form an equilateral triangle, and it can work because when two side lengths are added together, they are larger than the third side. What are the Symbols Used in Triangle Inequalities?The math symbols used in triangle inequalities are: greater than (>), less than (<), greater than or equal (≥), less than or equal (≤), and the not equal symbol (≠). What are the 3 Properties of the Triangle Inequality Theorem?The 3 properties of the triangle inequality theorem are:
A triangle is a three-sided polygon. It has three sides and three angles. The three sides and three angles share an important relationship. In Mathematics, the term “inequality” represents the meaning “not equal”. Let us consider a simple example if the expressions in the equations are not equal, we can say it as inequality. In this article, let us discuss what is triangle inequality in Maths, activities for explaining the concept of the triangle inequality theorem, and so on. In Mathematics, the term “triangle inequality” is meant for any triangles. Let us take a, b, and c are the lengths of the three sides of a triangle, in which no side is being greater than the side c, then the triangle inequality states that, c ≤ a+b This states that the sum of any two sides of a triangle is greater than or equal to the third side of a triangle. Let us discuss the relationship and equality and inequality of triangle, through an activity. Activity 1: On a paper mark two points Y and Z and join them to form a straight line. Mark another point X outside the line lying on the same plane of the paper. Join XY as shown. Now mark another point X’ on the line segment XY, join X’Z. Similarly, mark X’’ and join X’’Z with dotted lines as shown. From the above figure we can easily deduce that if we keep on decreasing the length of side XY such that XY> X’Y> X’’Y> X’’’Y the angle opposite to side XY also decreases i.e. ∠XZY >∠X’ZY >∠X’’ZY >∠X’’’ZY. Thus, from the above activity, we can infer that if we keep on increasing one side of a triangle then the angle opposite to it increases. Now let us try out another activity. Activity: Draw 3 scalene triangles on a sheet of paper as shown. Let us consider fig. (i). In ∆ABC, AC is the longest side and AB is the shortest. We observe that ∠B is the largest in measure and ∠C is the smallest. Similarly, in ∆XYZ, XY is the largest side and XZ is the smallest and ∠Z is the largest in measure and ∠Y is the smallest. In the last figure also the same kind of pattern is followed i.e. side PR is largest and so is the ∠Q opposite to it. Triangle Inequality TheoremLet us consider the triangle. The following are the triangle inequality theorems. Theorem 1: In a triangle, the side opposite to the largest side is greatest in measure. The converse of the above theorem is also true according to which in a triangle the side opposite to a greater angle is the longest side of the triangle. In the above fig., since AC is the longest side, the largest angle in the triangle is ∠B. Another theorem which follows can be stated as: Theorem 2: The sum of lengths of any two sides of a triangle is greater than the length of its third side. According to triangle inequality, AB + BC > AC. Video Lesson on BPT and Similar TrianglesTo learn more about triangle inequality proof and other properties please download BYJU’S- The Learning App.
Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin! Select the correct answer and click on the “Finish” button
Visit BYJU’S for all Maths related queries and study materials
out of 0 arewrong out of 0 are correct out of 0 are Unattempted View Quiz Answers and Analysis |
What is triangle inequality Class 7?
zusammenhängende Posts
Urheberrechte © © 2024 frojeostern Inc.
