1. **Problem statement:** We have an isosceles triangle $\triangle FOR$ with $FO = 7.0$ units, $FR = 10.0$ units, and $OR = 11.2$ units. Segment $OR$ is parallel to segment $AN$. We need to find the length of $AN$. 2. **Key information and formula:** Since $OR \parallel AN$, triangles $FOR$ and $FAN$ are similar by the AA (Angle-Angle) similarity criterion. 3. **Similarity ratio:** The sides of similar triangles are proportional. We know $OR = 11.2$ units and want to find $AN$. Let the length of $AN$ be $x$. 4. **Using the proportionality:** The corresponding sides $OR$ and $AN$ satisfy $$\frac{OR}{AN} = \frac{FR}{FN} = \frac{FO}{FA}$$ Since $OR \parallel AN$, the ratio of $OR$ to $AN$ equals the ratio of $FR$ to $FN$. 5. **Isosceles triangle property:** $\triangle FOR$ is isosceles, so two sides are equal. Given $FO = 7.0$ and $FR = 10.0$, the equal sides are likely $FO$ and $FR$ or $FO$ and $OR$. Since $FO \neq FR$, the equal sides are $FO$ and $OR$ or $FR$ and $OR$. But $OR$ is parallel to $AN$, so the isosceles property applies to $FO$ and $FR$. 6. **Calculate $AN$ using similarity:** Since $OR$ is parallel to $AN$, and $OR = 11.2$, $AN$ corresponds to $OR$ in the larger triangle. The scale factor from $\triangle FOR$ to $\triangle FAN$ is $\frac{AN}{OR} = \frac{x}{11.2}$. 7. **Using the side lengths:** The side $FR = 10.0$ corresponds to $FN$ in the larger triangle. Since $FN$ is longer than $FR$, the scale factor is greater than 1. 8. **Calculate scale factor:** The scale factor is $\frac{x}{11.2} = \frac{FN}{FR}$. But $FN = FR + RN$, and $RN$ is unknown. However, since $OR \parallel AN$, the triangles are similar, so the scale factor applies to all sides. 9. **Using the given lengths:** The scale factor is $\frac{x}{11.2} = \frac{AN}{OR}$. Since $AN$ is the side we want, and $OR = 11.2$, we can write $$x = 11.2 \times \frac{FA}{FO}$$ But $FA$ and $FO$ are unknown. Since $FO = 7.0$, and $FR = 10.0$, and $\triangle FOR$ is isosceles, the equal sides are $FO$ and $FR$ or $FO$ and $OR$. Given the data, the equal sides are $FO$ and $FR$. 10. **Conclusion:** Since $FO = 7.0$ and $FR = 10.0$ are not equal, the isosceles triangle must have $FO = FR$ or $FO = OR$. Given the problem states $\triangle FOR$ is isosceles, the equal sides are $FO$ and $FR$, so $FO = FR = 10.0$ or $7.0$? Since given $FO=7.0$ and $FR=10.0$, this contradicts isosceles unless the problem has a typo. Assuming $FO = FR = 10.0$ (correcting the typo), then the scale factor is $$\frac{AN}{OR} = \frac{FR}{FO} = \frac{10.0}{7.0} = 1.42857$$ Therefore, $$AN = OR \times 1.42857 = 11.2 \times 1.42857 = 16.0$$ 11. **Final answer:** $AN = 16.00$ units (rounded to nearest hundredth).