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The Triangle Inequality ($|x+y|\leq|x|+|y|$) is useful later on in the student's math education (e.g. in proving results about limits).

But for the high-school student, are there any useful and interesting applications of the Triangle Inequality?

Please exclude from your answers any use of the Triangle Inequality to simply derive some similar inequality (e.g. $||x|-|y||\leq |x+y|$), unless that derived inequality can be shown to have some interesting applications (other than further deriving other similar inequalities).

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At its most basic, the Triangle Inequality says that it is (in general) a shorter trip to go directly from point $A$ to point $B$ than it would be to stop at point $C$ in between. That is, the length of any one side of a triangle is less than the sum of the lengths of the other two sides. That is the source of the name "Triangle Inequality", and is the most elementary version of the (more complicated) claim that a straight line is the shortest distance between two points. This is certainly a "useful" application, although whether or not it counts as "interesting" is, I suppose, debatable.

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  • $\begingroup$ I think the Triangle Inequality is so-called because in two or higher dimensions, we have $|\mathbf{x}+\mathbf{y}|\leq|\mathbf{x}|+|\mathbf{y}|$, where we can indeed apply the "triangle" or "shortest distance" interpretation. I'm not sure though if this interpretation applies to $|x+y|\leq|x|+|y|$ (one-dimensional case), where $x$ and $y$ are instead just real numbers? Related. $\endgroup$
    – user18187
    Feb 22 at 1:06
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    $\begingroup$ @user24096 Certainly! Let $A, B$ and $C$ be real numbers, interpreted as points on a number line. Then $x = B - A$ is the distance from $A$$ to $B$$, and so forth. The triangle inequality still says that going directly from $A$ to $B$ is always a distance less than or equal to going from $A$ to $C$ to $B$, with equality holding only if $C$ is in between $A$ and $B$. $\endgroup$
    – mweiss
    Feb 22 at 19:01
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There is the broken stick problem, which asks for the probability that if you break a stick at two points chosen uniformly at random along its length, then the resulting pieces can be used to form a triangle. This is an easily stated problem that is not so trivial to solve.

Admittedly, the hard part of this problem isn't the triangle inequality, but the triangle inequality is the first crucial step in the solution. I think that for many students, the triangle inequality looks so trivial that it doesn't seem like it could be useful. But when students first see the broken stick problem, they often don't immediately recognize that the triangle inequality is relevant. It takes a bit of thought. This thought process helps them appreciate why the triangle inequality is worth stating formally.

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    $\begingroup$ Inspired by your recent answer to another question, which recommends something that should have occurred to me yesterday when I posted some literature comments, I'll mention that science-oriented high school students might find it interesting to learn that, from a mathematical standpoint, Heisenberg's Uncertainty Principle is a direct application of a generalization of the triangle inequality. $\endgroup$
    – Dave L Renfro
    Feb 24 at 11:20
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Here are two examples (one for your one-dimensional question and one in 2D, which is not really what you are asking about)

  • Consider the following (not really lifelike) situation. There are two shops along a straight road and we are considering building a bakery somewhere on the road that supplies bread to these shops. The delivery van can only carry enough bread for one shop. Where shall we build the bakery to minimise the distance travelled by the delivery van, if the van starts and finishes at the bakery? The answer is: anywhere between the two shops. This can be justified using common sense or using coordinates on the number line and the triangle inequality you mentioned. An interesting extension is to discuss this problem with 3, 4, 5, ... shops (still along a straight road). Note that these questions can be also answered using graphs involving sums of shifted absolute value functions, which can also be interesting to investigate with high school students.
  • I know that the following example is to illustrate the geometric triangle inequality (which is not what you were asking about), but it still might be interesting to mention to high school students when you discuss why your algebraic expression is called triangle inequality. It is Heron's shortest distance problem (which can be illustrated on a pool table): What is the shortest path from A to B that touches a given line? Of course the interesting part of the question is when the two points are on the same side of the line. Here is a link to an activity related to this.
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