1. **State the problem:** We need to find the length of segment $LW$ in a right triangle $LYW$ where $LY = 3.5$ units and $AY = 7.0$ units, with $A$ on $LY$ and $AY$ perpendicular to $LY$. 2. **Identify similar triangles:** Since $AY$ is perpendicular to $LY$, triangles $LAY$ and $WYA$ are right triangles sharing angle $Y$, so they are similar by AA similarity. 3. **Set up the proportion:** Similar triangles have proportional sides. Using corresponding sides, $$\frac{LW}{AY} = \frac{LY}{LW}$$ 4. **Substitute known values:** $LY = 3.5$, $AY = 7.0$, so $$\frac{LW}{7.0} = \frac{3.5}{LW}$$ 5. **Solve for $LW$:** Cross-multiply: $$LW^2 = 3.5 \times 7.0$$ $$LW^2 = 24.5$$ 6. **Calculate $LW$:** $$LW = \sqrt{24.5} \approx 4.95$$ 7. **Final answer:** The length of $LW$ is approximately **4.95 units**.