1. **Problem Statement:** Given triangle $\triangle JKL$ with incenter $P$, and the lengths $PO=31$, $KM=34$, and $PL=48$, find the lengths of segments $PM$, $KN$, $OL$, and $KP$. 2. **Understanding the Incenter and Angle Bisectors:** The incenter $P$ is the point where the angle bisectors of the triangle intersect. It is equidistant from all sides of the triangle. 3. **Given Lengths and What They Represent:** - $PO=31$ is the distance from $P$ to point $O$ on segment $JL$. - $KM=34$ is a segment on side $JK$. - $PL=48$ is the length from $P$ to vertex $L$. - $PM=31$ and $KN=34$ are given as well. 4. **Finding $PM$:** Given directly as $PM=31$. 5. **Finding $KN$:** Given directly as $KN=34$. 6. **Finding $OL$:** Since $O$ lies on $JL$ and $P$ is the incenter, $PO$ is perpendicular to $JL$ and $OL$ is the remaining segment on $JL$ from $O$ to $L$. 7. **Finding $KP$:** Since $KM=34$ and $P$ lies on the angle bisector from $K$, $KP$ is part of $KM$. 8. **Summary of Known Values:** - $PM=31$ - $KN=34$ - $PO=31$ - $PL=48$ - $KM=34$ 9. **Final Answers:** - $PM=31$ - $KN=34$ - $OL$ cannot be determined with given data. - $KP$ cannot be determined with given data. Note: Without additional information or relationships, $OL$ and $KP$ cannot be uniquely found.