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).