Triangle inequality theorem: the sum of any two sides of a triangle is greater than or equal to the third side.
Given lengths:
1-
[tex]\begin{gathered} 17+4\ge8 \\ 21\ge8 \\ \\ 4+8\ge17 \\ 12\ge17 \\ \\ 17+8\ge4 \\ 25\ge4 \end{gathered}[/tex]
As you can see above not all the inequalities are true. Then, those lengths cannot be side lengths of a triangle.
2-
[tex]\begin{gathered} 13.3+4.3\ge9.9 \\ 17.6\ge9.9 \\ \\ 4.3+9.9\ge13.3 \\ 14.2\ge13.3 \\ \\ 9.9+13.3\ge4.3 \\ 23.2\ge4.3 \end{gathered}[/tex]
All the inequalities are true. Then, those lengths can be side lengths of a triangle
3-
[tex]\begin{gathered} 22+6\ge6 \\ 28\ge6 \\ \\ 6+6\ge22 \\ 12\ge22 \\ \end{gathered}[/tex]
As you can see above not all the inequalities are true. Then, those lengths cannot be side lengths of a triangle
4-
[tex]\begin{gathered} 13+6\ge16 \\ 19\ge16 \\ \\ 6+16\ge13 \\ 22\ge13 \\ \\ 13+16\ge6 \\ 29\ge6 \end{gathered}[/tex]
All the inequalities are true. Then, those lengths can be side lengths of a triangle
