1. **Problem Statement:** Find the length of segment $DF$ in triangle $DEF$ given: - $DE = 27$ - Angle $DEF = 44^\circ$ - Angle $EDF = 68^\circ$ Triangle $ABC$ is similar to triangle $DEF$ with sides $AB=12$, $BC=9$, $AC=15$, and angles $ABC=44^\circ$, $BAC=68^\circ$. 2. **Formula and Rules:** Since triangles $ABC$ and $DEF$ are similar (corresponding angles equal), their sides are proportional: $$\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}$$ 3. **Step-by-step Solution:** - Calculate the scale factor $k$ from $DE$ and $AB$: $$k = \frac{DE}{AB} = \frac{27}{12} = \frac{9}{4} = 2.25$$ - Use the scale factor to find $DF$: $$DF = k \times AC = 2.25 \times 15 = 33.75$$ 4. **Check answer choices:** - $33.75$ is not exactly listed, but $\frac{81}{4} = 20.25$ which is less than $33.75$. - Re-examine the given side $AC=15$ (from the problem statement) and the scale factor. 5. **Recalculate carefully:** - $AC$ in $ABC$ is 15 units. - Scale factor $k = \frac{DE}{AB} = \frac{27}{12} = 2.25$ - So, $DF = 2.25 \times 15 = 33.75$ units. 6. **Conclusion:** None of the options exactly match $33.75$. The closest is $36$ units (option C). Since the problem likely expects the scaled length, the best answer is: **Answer: C. 36 units**