Triangle Inequality Theorem (Video, Proofs, Examples) (2024)

Triangle inequality

Trianglesare 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 a3−4−5triangle, 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 thetriangle inequality theorem, which helps you determine quickly if a set of three numbers could be used to construct a triangle.

Triangle inequality theorem

TheTriangle Inequality Theoremstates that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Consider our3−4−5triangle 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 the3−4−8line 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

TheTriangle Inequality Theoremis easy to prove. You can prove it yourself with a piece of paper, a ruler, and a pencil. Draw a15cmline on the paper. From one endpoint, draw a7cmline at any upward angle you please.

Measure from that line's endpoint to the far end of the15cmline. That measure, no matter the angle you created, will be greater than15cm.

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 theTriangle Inequality Theoremis 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 PointTto PointRis, let's say,100meters, with the distance from PointRto PointY, let's say,250meters.

If this were a triangle, that means "side"TYwould be350meters. If you want the triangle to have any kind of interior angles at all, though, either "side"TRor "side" RY has to be greater than the collinear distances, so PointRwould no longer be on the line segmentTY.

Say you increase sideTRto210meters. That moves PointRaway from the line, like this:

[same drawing, but PointRis now above and away from line segment/side TY]

To have line segment and sideRYmeet the new end ofTR, sideRYmust be280meters:

[new drawing of △TRY with sides labeled: TR 210 m; RY 280 m; TY still 350 m]

Now if you add up sidesTRandRY, you get490m, not350m.

Aha, you say, what if we swingTRall 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 fromTYis still350mandTR(orRT; it's the same line segment in either direction) is still100m.

Now, though,RYis the "long side" of our degenerate triangle, at450m. 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 upRYso PointRis no longer collinear with PointsTandY. 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.