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Home » Triangles » Triangle Inequalities » Reverse Triangle Inequality Theorem

Reverse Triangle Inequality Theorem

Last updated: May 1, 2024 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In this problem we will prove the Reverse Triangle Inequality Theorem, using what we have already proven In a previous problem- the Triangle Inequality.

The Triangle Inequality theorem states that in a triangle, the sum of the lengths of any two sides is larger than the length of the third side.

The Reverse Triangle Inequality states that in a triangle, the difference between the lengths of any two sides is smaller than the third side. Or stated differently, any side of a triangle is larger than the difference between the two other sides.

In addition to formally proving that theorem, we also provided an intuitive explanation of why it is true, using what we know about the shortest distance between two points - it is the straight line connecting them.

We can think of any triangle as being composed of one side, which is the straight line between two points. The other two side are another path between those two points. But that path goes through a third point. Since this is not the straight line, going through that path it is going to be longer. So the sum of the two sides which form the alternate path is longer than the straight line.

Here, we will do the same - we will provide a formal proof of the theorem, and then give an intuitive explanation of why it is true.

Problem

Show that |AB|>||AC|-|CB||, |AC|>||AB|-|CB|| and |BC|>||AB|-|AC||

Strategy

We will use the Triangle Inequality Theorem we have already proven, and do a little manipulation of the lengths, using basic algebra to get the desired result.

Proof

(1) |AB| + |AC| > |CB| //Triangle Inequality Theorem
(2) |AB| + |AC| -|AC| > |CB|-|AC| //(1) Subtracting the same quantity from both side maintains the inequality
(3) |AB| > |CB| - |AC| = ||AC|-|CB|| //(2), properties of absolute value

And similarly for the other two sides:

(4) |AB| + |AC| - |AB| > |CB|-|AB| // (1) Subtracting the same quantity from both side maintains the inequality
(5) |AC| > |CB|-|AB| = ||AB|-|CB|| //(4), properties of absolute value

(6) |AC|+|BC| > |AB| //Triangle Inequality Theorem
(7) |AC| + |BC| -|AC| > |AB|-|AC| //(6) Subtracting the same quantity from both side maintains the inequality
(8) |BC| > |AB| - |AC| = ||AB|-|AC|| //(7), properties of absolute value

Intuitive explanation

So, now that we have formally proven the Reverse Triangle Inequality, let's try to intuitively explain why this is so:

Imagine that you walk from point A to point B, and that is one side of a triangle. Then you walk from point B to C, in any direction. BC is the second side of the triangle. Because the shortest distance between two points is a straight line, the closest you could be to point A is if you traced your steps directly back toward A, on the same line, to point D.

the difference between two sides of a triangle

BD is the same length as BC, so AD is the difference between the two sides AB And BC. It is also smaller than AC, because D is the closest you could be to A after walking from B. But AC is the third side of the triangle, and is larger than AD, which is the difference between the two sides.

« Find the area of a parallelogram using diagonals
Parallel Lines are Equidistant »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected].

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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