Triangle Inequality Theorem
Triangle inequality
Triangles are the simplest polygons, composed of only three sides, or line segments. Those three line segments cannot be just any random lengths, though. Only particular numbers can work, like a 3−4−5 triangle, with sides 3 units, 4 units and 5 units long.
If you change just one of those numbers, and you cannot make the triangle. Change the 5 units side to an 8 unit side, and you can immediately understand that the 8-meter-long line segment is past the reach of the other two line segments.
The shortest distance between two points is a straight line. That is the heart of the triangle inequality theorem, which helps you determine quickly if a set of three numbers could be used to construct a triangle.
Triangle inequality theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Consider our 3−4−5 triangle example above. Add up any two sides of it. The sum of any of those two sides must be greater than the remaining side:
Compare that to the 3−4−8 line segments above:
The last inequality shows that the three line segments cannot form a triangle, since the two sides together are shorter than the remaining side.
Triangle inequality theorem proofs
The Triangle Inequality Theorem is easy to prove. You can prove it yourself with a piece of paper, a ruler, and a pencil. Draw a 15 cm line on the paper. From one endpoint, draw a 7 cm line at any upward angle you please.
Measure from that line's endpoint to the far end of the 15 cm line. That measure, no matter the angle you created, will be greater than 15 cm.
Another proof is found in looking at just about any map. Find a straight road on a map, and a road branching off from it, connecting to another road that joins back up to the first road.
Looking at the map, you should notice that taking the two short sections of road will still make you drive more than just driving down the first, straight road.
Side lengths of a triangle
Yet another way to prove the Triangle Inequality Theorem is by reducing the argument about the lengths of the sides to an absurd level, by creating a degenerate triangle.
In geometry, a degenerate triangle is a set of three points on a line, like this:
[insert drawing line segment TY with Point R between but not centered; this proof may be better animated than drawn]
These points are considered collinear (all in a line). The distance from Point T to Point R is, let's say, 100 meters, with the distance from Point R to Point Y, let's say, 250 meters.
If this were a triangle, that means "side" TY would be 350 meters. If you want the triangle to have any kind of interior angles at all, though, either "side" TR or "side" RY has to be greater than the collinear distances, so Point R would no longer be on the line segment TY.
Say you increase side TR to 210 meters. That moves Point R away from the line, like this:
[same drawing, but Point R is now above and away from line segment/side TY]
To have line segment and side RY meet the new end of TR, side RY must be 280 meters:
[new drawing of △TRY with sides labeled: TR 210 m; RY 280 m; TY still 350 m]
Now if you add up sides TR and RY, you get 490 m, not 350 m.
In order for the triangle to actually be a triangle, any two sides must be greater than the length of the remaining side. When they are equal to the length of the remaining side, you have a straight line.
Aha, you say, what if we swing TR all the way in the other direction, opening up the angle completely?
Now you have the points along the line segment in this order: RTY, and the distance from TY is still 350 m and TR (or RT; it's the same line segment in either direction) is still 100 m.
Now, though, RY is the "long side" of our degenerate triangle, at 450 m. You still have three collinear points, so you still have no proper triangle.
The only way to get a recognizable triangle in this configuration is to shorten up RY so Point R is no longer collinear with Points T and Y. So, once again, any two sides will add to a number greater than the third side.
Lesson summary
Now that you worked through this lesson, you are able to recall and state the triangle inequality theorem, explain why the triangle inequality theorem is true, and apply the triangle inequality theorem to line segments to test to see if they can form a triangle.