Which three lengths could be the lengths of the sides of a triangle?

Solution:

Let us see the nature of the sides of a triangle using the triangle inequality theorem.

The two conditions of the sides of a triangle are:

1. The sum of the two sides is always greater than the third side.
   • a + b > c
   • b + c > a
   • c + a > b
2.  The difference between the two sides is less than the third side.
   • |a - b| < c
   • |b - c| < a
   • |c - a| < b

Hence, three lengths could be the lengths of the sides of a triangle if and only if the sum of two sides is always greater than the third side and the difference of the two sides is less than the third side.

Which three lengths could be the lengths of the sides of a triangle?

Summary:

(a, b, c) can be the sides of a triangle if and only if the sum of two sides is always greater than the third side and the difference of the two sides is less than the third side.