1. **Stating the problem:** Triangles $XYZ$ and $PQR$ are similar. Given sides $XY=35$, $YZ=30$, $XZ=X$, and corresponding sides $PQ=28$, $QR=12$, $PR=n$, find $X$ and $n$. 2. **Formula and rules:** For similar triangles, corresponding sides are proportional: $$\frac{XY}{PQ} = \frac{YZ}{QR} = \frac{XZ}{PR}$$ 3. **Calculate the scale factor:** $$\frac{XY}{PQ} = \frac{35}{28} = \frac{5}{4}$$ $$\frac{YZ}{QR} = \frac{30}{12} = \frac{5}{2}$$ Since these are not equal, check if sides correspond differently. Assume $XY$ corresponds to $PQ$, $YZ$ to $QR$, and $XZ$ to $PR$. 4. **Check proportionality:** $$\frac{35}{28} = \frac{5}{4}$$ $$\frac{30}{12} = \frac{5}{2}$$ These are not equal, so the given sides may correspond differently. Try matching $XY$ to $QR$ and $YZ$ to $PQ$: $$\frac{XY}{QR} = \frac{35}{12}$$ $$\frac{YZ}{PQ} = \frac{30}{28} = \frac{15}{14}$$ Not equal either. 5. **Try $XY$ to $PR$ and $YZ$ to $PQ$:** $$\frac{XY}{PR} = \frac{35}{n}$$ $$\frac{YZ}{PQ} = \frac{30}{28} = \frac{15}{14}$$ We don't know $n$ yet, so use the third ratio: $$\frac{XZ}{QR} = \frac{X}{12}$$ 6. **Set ratios equal:** $$\frac{35}{n} = \frac{15}{14}$$ Solve for $n$: $$35 \times 14 = 15 \times n$$ $$490 = 15n$$ $$n = \frac{490}{15} = \frac{98}{3} \approx 32.67$$ 7. **Find $X$ using the ratio:** $$\frac{X}{12} = \frac{15}{14}$$ $$X = 12 \times \frac{15}{14} = \frac{180}{14} = \frac{90}{7} \approx 12.86$$ **Final answers:** $$X = \frac{90}{7}$$ $$n = \frac{98}{3}$$