1. **Problem statement:** Calculate the length $x$ in the triangle where $MN \parallel BC$ and given lengths are $BM=5$, $AN=4$, and $BC=10$. 2. **Formula and theorem used:** According to Thales' theorem, if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Thus, we have: $$\frac{AM}{AB} = \frac{AN}{AC} = \frac{MN}{BC}$$ 3. **Apply the theorem:** Here, $MN \parallel BC$, so: $$\frac{BM}{AB} = \frac{AN}{AC}$$ Given $BM=5$, $BC=10$, and $AN=4$, we need to find $x=MN$. 4. **Calculate $AB$ and $AC$:** Since $BM=5$ and $MN$ lies between $M$ and $N$, and $BC=10$, we can write: $$AB = BM + AM$$ But $AM$ is unknown, so we use the proportionality: $$\frac{BM}{AB} = \frac{AN}{AC}$$ 5. **Express $x$ in terms of known lengths:** Since $MN \parallel BC$, the ratio of $MN$ to $BC$ equals the ratio of $AM$ to $AB$: $$\frac{MN}{BC} = \frac{AM}{AB}$$ 6. **Calculate $x$:** Given $BC=10$, and $MN$ is the segment between $M$ and $N$, the length $x$ corresponds to $MN$. Since $BM=5$ and $BC=10$, $MC=5$. Using the proportionality: $$\frac{BM}{AB} = \frac{AN}{AC}$$ But $AN=4$ and $AC$ is unknown. Assuming $AB=BM + AM = 5 + x$ and $AC=AN + NC = 4 + y$ (where $y$ is unknown), we need more information to solve for $x$. **Note:** The problem as stated lacks sufficient data to find $x$ directly without additional lengths or relationships. **Final answer:** Cannot determine $x$ with given data alone.