1. **State the problem:** We are given the side lengths of triangle $NOP$ as $PN=68$ and $ON=55.2$. We need to find the perimeter of triangle $NOP$, which is the sum of all its sides: $NO + OP + PN$. 2. **Identify the missing side:** The side $OP$ is not given directly. However, from the problem, we only have $PN$ and $ON$ for triangle $NOP$. Since $OP$ is not given, we assume the problem expects the perimeter as the sum of the given sides plus the missing side $OP$. 3. **Check if $OP$ is given or can be inferred:** The problem does not provide $OP$ length. Since the problem only gives $PN$ and $ON$, and asks for the perimeter of $NOP$, we assume $OP$ is the side $LM$ or $MK$ or $LK$ from the other triangle, but since the triangles are separate, we only sum the given sides of $NOP$. 4. **Calculate the perimeter:** The perimeter $P$ of triangle $NOP$ is $$P = NO + OP + PN$$ Given $NO = 55.2$, $PN = 68$, and $OP$ is missing, so we cannot calculate the exact perimeter without $OP$. 5. **Conclusion:** Since $OP$ is not given, the problem likely expects the perimeter as the sum of the given sides $PN$ and $ON$ plus the missing side $OP$ which is not provided. Therefore, the perimeter cannot be calculated exactly with the given data. **Final answer:** Cannot determine the perimeter of triangle $NOP$ with the given information.