1. **Stating the problem:** We need to find the length of segment $JN$ by using the similarity of two right triangles $\triangle JKL$ and $\triangle JNM$. 2. **Identifying similar triangles:** Both triangles have a right angle ($\angle L$ in $\triangle JKL$ and $\angle M$ in $\triangle JNM$) and share angle $J$, so by angle-angle similarity, $\triangle JKL \sim \triangle JNM$. 3. **Setting up the proportion:** Corresponding sides of similar triangles are proportional. We set up the proportion: $$\frac{JN}{JK} = \frac{JM}{JL}$$ where $JN$ corresponds to $JK$, and $JM$ corresponds to $JL$. 4. **Substituting known lengths:** Given that $JK = JN + 6.9$, $JM = 4.4$, and $JL = 4.4 + 3.6 = 8.0$, the proportion becomes: $$\frac{JN}{JN + 6.9} = \frac{4.4}{8.0}$$ 5. **Solving the proportion:** Multiply both sides by $JN + 6.9$: $$JN = \frac{4.4}{8.0} (JN + 6.9)$$ Multiply both sides by 8.0 to clear the denominator: $$8.0 JN = 4.4 (JN + 6.9)$$ Distribute 4.4: $$8.0 JN = 4.4 JN + 30.36$$ Subtract $4.4 JN$ from both sides: $$3.6 JN = 30.36$$ Divide both sides by 3.6: $$JN = \frac{30.36}{3.6} \approx 8.4$$ 6. **Conclusion:** The length of segment $JN$ is approximately $8.4$ units.