1. **Problem statement:** We have triangle $\triangle DEF$ with medians $DK$, $EL$, and $FJ$ intersecting at centroid $M$. Given lengths are $ML=9$, $MJ=8$, and $DK=24$. We need to find lengths $FJ$, $DM$, and $EM$. 2. **Key property of centroid:** The centroid divides each median into a ratio of $2:1$, with the longer segment between the vertex and the centroid. 3. **Using the centroid property on median $EL$:** Since $ML=9$ is the shorter segment, $$EL = 3 \times ML = 3 \times 9 = 27$$ 4. **Using the centroid property on median $FJ$:** Since $MJ=8$ is the shorter segment, $$FJ = 3 \times MJ = 3 \times 8 = 24$$ 5. **Using the centroid property on median $DK$:** Given $DK=24$, the centroid divides it as $DM:MK = 2:1$. 6. **Calculate $DM$:** $$DM = \frac{2}{3} \times DK = \frac{2}{3} \times 24 = 16$$ 7. **Calculate $EM$:** Since $EL=27$, and $EM$ is the longer segment, $$EM = \frac{2}{3} \times EL = \frac{2}{3} \times 27 = 18$$ **Final answers:** - $FJ = 24$ - $DM = 16$ - $EM = 18$