Triangle Inequality: Can Three Lengths Make a Triangle?

Start with what a side is. Each side of a triangle is the straight path between two corners, and a straight path is the shortest way to get from one point to another.

Now think about the longest side. The other two sides connect its two ends as well, but they go the long way around, through the third corner. That route leaves the straight path, so it is a detour, and a detour is longer than the straight path. So in a true triangle, the two shorter sides together must be longer than the longest side.

The table below puts numbers on this idea. The longest side is 10 cm in every row, and only the two shorter sides change. The first row is the opening question.

Why is one comparison enough? The detour idea is true for every side, not only the longest. So in a nondegenerate triangle, any two sides together are longer than the third side. That gives three comparisons: $a + b > c$, $a + c > b$ and $b + c > a$.

When the lengths are positive and named so that $a \le b \le c$, two of these are true every time:

• $a + c > b$, because $c$ alone is at least as long as $b$, and adding the positive length $a$ makes the sum greater than $b$.
• $b + c > a$, because $c$ alone is at least as long as $a$, and adding the positive length $b$ makes the sum greater than $a$.

So $a + b > c$ is the only comparison that can fail. That is why you add the two shorter lengths. If two lengths tie for longest, either one can be $c$, and the test works the same way.

Takeaway: To decide whether three positive lengths in the same unit form a nondegenerate triangle, add the two shorter lengths and compare the sum with the longest. If the sum is more than the longest, a triangle is possible. If the sum equals the longest, the sides lie flat in a straight line. If the sum is less, the two shorter sides cannot meet. In those last two cases, there is no triangle.

Next, you will draw triangles from three given side lengths and from given angle measures.