Demystifying Triangle Inequality

This is the first part of  the Triangle Inequality Series.

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Given three segments, how are we sure that they will form a triangle?

Will the segments with lengths 3, 4, and 8 units form a triangle? What about 3, 5, and 8?

What conditions with respect to segment lengths must be satisfied, so that three segments form a triangle?

Shown below are the segments with lengths 3, 4 and 8 units. In the following illustrations, squares are used to clearly indicate the segments’ lengths.  Each square has side 1 unit.  As we can see, we cannot form a triangle if the sum of the lengths of the two sides is less than the third side.

Let us now examine a triangle with side lengths 3, 5, and 8 units. Referring to Figure 2, the sides will only be coinciding and will never meet when rotated outward as shown.

If the sum of the lengths of the two sides is greater than the third side, then a triangle can be formed as shown in Figure 3.

In general, to form a triangle using three segments, the sum of the two side lengths must be greater than the third side length. If the triangle has side lengths a, b and c, the following conditions must be satisfied: (1) a + b > c, (2) b + c > a, and (3) a + c > b. This theorem is called triangle inequality.

Generally speaking, triangle inequality states that the length of the two sides of a triangle should always be greater than length of the third side. Click the picture below to explore the GeoGebra applet about triangle inequality. Update: David Travis has an excellent applet modeled from the drawings above.

Exercise: A triangle has side lengths 5, 7, x. Find all the possible integral values of x.

To be concluded.