Algor Cards

Triangle Inequalities and Their Applications

Concept Map

Algorino

Edit available

Triangle inequalities form the basis for understanding the relationships between the sides and angles of triangles. The Triangle Inequality Theorem states that the sum of any two sides of a triangle must exceed the third side's length. This principle, along with the Angle-Side Relationship, which correlates the size of an angle with the length of its opposite side, and the Exterior Angle Inequality Theorem, are essential for geometric reasoning and solving problems when dealing with incomplete data about triangles.

Principles of Triangle Inequalities

Triangle inequalities are essential rules that govern the relationships between the sides of any triangle, whether it is scalene, isosceles, or equilateral. The Triangle Inequality Theorem is a cornerstone of these principles, stating that the sum of the lengths of any two sides of a triangle must exceed the length of the third side. This theorem can be expressed with the inequality |AC| + |BC| > |AB|, and similarly for the other pairs of sides, where |XY| represents the length of the side connecting vertices X and Y. Understanding these inequalities is crucial for comprehending the possible dimensions and shapes of triangles.
Scalene triangle formed by three wooden rulers with a blue, red, and yellow paper triangle at each corner on a light surface.

Geometric Proof of the Triangle Inequality Theorem

The Triangle Inequality Theorem can be proven using a straightforward geometric argument. Consider a triangle with vertices A, B, and C. By constructing arcs with centers at A and B and radii equal to the lengths of sides AC and BC, respectively, it becomes apparent that the direct path along side AB must be shorter than the path that travels along the two arcs. This is because any detour from the direct path, such as traveling along the arcs, will result in a longer total distance. This geometric visualization confirms that the sum of the lengths of two sides of a triangle is always greater than the length of the third side.

Show More

Want to create maps from your material?

Enter text, upload a photo, or audio to Algor. In a few seconds, Algorino will transform it into a conceptual map, summary, and much more!

Learn with Algor Education flashcards

Click on each Card to learn more about the topic

00

Triangle Inequality Theorem Expression

|AC| + |BC| > |AB|; |AB| + |BC| > |AC|; |AB| + |AC| > |BC|.

01

Applicability of Triangle Inequalities

Applies to all triangles: scalene, isosceles, equilateral.

02

Purpose of Triangle Inequalities

Determines possible dimensions, ensures triangle formation.

Q&A

Here's a list of frequently asked questions on this topic

Can't find what you were looking for?

Search for a topic by entering a phrase or keyword