Question

If LM=18 cm and MN=5 cm, then what are the possible lengths for LN so that LM, MN, and NL can form a triangle?

Answer 1

`\overline{LM}+\overline{MN}>\overline{NL}\\ \overline{LM}+\overline{NL}>\overline{MN}\\ \overline{NL}+\overline{MN}>\overline{LM}\\\\ 18+5>\overline{NL}\\ \overline{NL}<23\\\\ 18+\overline{NL}>5\\ \overline{NL}>-13\\\\ \overline{NL}+5>18\\ \overline{NL}>13\\\\ \overline{NL}<23\wedge \overline{NL}>-13 \wedge \overline{NL}>13\\ \boxed{\overline{NL}\in(13,23)}`

Answer 2

The rule for a triangles sides is that the sum of two sides must be greater than the other side. The difference of two sides must be less than the other side.
Our two lengths are 18 and 5
18+5=23
18-5=13
LN must be between 13 and 23