1. **State the problem:** Given triangles $\triangle ART \sim \triangle FIN$ with altitudes $AS = 4$ and $FE = 3$, and sides $AT = x+1$ and $FN = x-1$, find the lengths of $AT$ and $FN$. 2. **Use the property of similar triangles:** Corresponding sides of similar triangles are proportional. Since $\triangle ART \sim \triangle FIN$, we have $$\frac{AT}{FN} = \frac{AS}{FE}$$ 3. **Substitute the known values:** $$\frac{x+1}{x-1} = \frac{4}{3}$$ 4. **Cross multiply to solve for $x$:** $$3(x+1) = 4(x-1)$$ 5. **Expand both sides:** $$3x + 3 = 4x - 4$$ 6. **Rearrange to isolate $x$:** $$3 + 4 = 4x - 3x$$ $$7 = x$$ 7. **Find $AT$ and $FN$ by substituting $x=7$:** $$AT = x + 1 = 7 + 1 = 8$$ $$FN = x - 1 = 7 - 1 = 6$$ **Final answer:** $AT = 8$ and $FN = 6$.