Triangle Inequality Theorem
The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.
Adjust the triangle by dragging the points A,B or C. Notice how the longest side is always shorter than the sum of the other two.
In the figure above, drag the point C up towards the line AB. As it gets closer you can see that the line AB is always
shorter than the sum of AC and BC. It gets close, but never quite makes it until C is actually on the line AB and the figure is no longer a triangle.
The shortest distance between two points is a straight line. The distance from A to B will always be longer if you have to 'detour' via C.
To illustrate this topic, we have picked one side in the figure above, but this property of triangles is always true no matter
which side you initially pick. Reshape the triangle above and convince yourself that this is so.
A triangle cannot be constructed from three line segments if any of them is longer than the sum of the other two.
Drag any orange dot. Notice you cannot make a triangle out of these three segments.
In the above figure, the lengths of the sides A and B add up to less than the length of C.
This violates the Triangle Inequality Theorem, and so it is not possible for the three lines segments to be made into a triangle.
In the figure above, drag both loose ends down on to the line segment C, to see why this is so.
[Thanks to Jeff Holcomb, Santa Cruz City Schools, California for the inspiration for the above applet]
Related triangle topics
Perimeter / Area
Congruence and Similarity
Triangle quizzes and exercises
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