1. **Problem Statement:** We are given an isosceles triangle $\triangle FOR$ with $\overline{OR} \parallel \overline{AN}$. We know the lengths $FO = 7.0$, $OR = 11.2$, and $FR = 10.0$. We need to find the length of $\overline{AN}$. 2. **Key Concept:** Since $\overline{OR} \parallel \overline{AN}$, triangles $\triangle FOR$ and $\triangle FAN$ are similar by the AA (Angle-Angle) similarity criterion. 3. **Similarity Ratios:** Corresponding sides of similar triangles are proportional. So, $$\frac{FO}{FA} = \frac{OR}{AN} = \frac{FR}{FN}$$ 4. **Using the Isosceles Property:** $\triangle FOR$ is isosceles with $FO = FR = 7.0$ or $10.0$? Given $FO=7.0$ and $FR=10.0$, the triangle is not isosceles by these sides, so the isosceles property likely applies to $\triangle FOR$ with equal sides $FO = FR$. Since given values differ, we assume the problem means $\triangle FOR$ is isosceles with $FO = FR = 10.0$ (assuming a typo or misinterpretation). We proceed with $FO = FR = 10.0$. 5. **Calculate Scale Factor:** Using the parallel lines and similarity, $$\frac{OR}{AN} = \frac{FO}{FA}$$ We know $OR = 11.2$, $FO = 7.0$ (given), but we need $FA$ and $AN$. Since $FA$ and $AN$ are parts of the larger triangle, and $OR$ is parallel to $AN$, the scale factor $k = \frac{AN}{OR} = \frac{FA}{FO}$. 6. **Find $AN$:** Since $OR$ corresponds to $AN$, and the scale factor is $k = \frac{AN}{OR}$, we can write: $$AN = k \times OR$$ But we need $k$. Using the isosceles property and given lengths, the scale factor is: $$k = \frac{FA}{FO}$$ Assuming $FA = FO + FR = 7.0 + 10.0 = 17.0$ (since $F$ to $A$ is the sum of segments $FO$ and $OR$ or $FR$), then $$k = \frac{17.0}{7.0} = 2.42857$$ 7. **Calculate $AN$:** $$AN = k \times OR = 2.42857 \times 11.2 = 27.2$$ 8. **Final Answer:** Rounded to the nearest hundredth, $$AN = 27.20 \text{ units}$$