Question: Find the perimeter of $\triangle DEF$. Round your answer to the nearest tenth if necessary. Figures are not necessarily drawn to scale. C $74^\circ$ 8 $57^\circ$ B 9 $49^\circ$ 10 A E $57^\circ$ 28 $49^\circ$ D 22.4 $74^\circ$ x F

1. **State the problem:** We need to find the perimeter of $\triangle DEF$, where two sides are given as $28$ and $22.4$, and the third side $x$ is unknown. The angles are $57^\circ$ at $E$, $49^\circ$ at $D$, and $74^\circ$ at $F$. 2. **Identify the knowns and unknowns:** - Side $DE = 22.4$ - Side $EF = 28$ - Side $DF = x$ (unknown) - Angles: $\angle E = 57^\circ$, $\angle D = 49^\circ$, $\angle F = 74^\circ$ 3. **Use the Law of Sines to find $x$: $$\frac{x}{\sin 57^\circ} = \frac{22.4}{\sin 74^\circ}$$** 4. **Solve for $x$: $$x = \frac{22.4 \times \sin 57^\circ}{\sin 74^\circ}$$** 5. **Calculate the sine values:** - $\sin 57^\circ \approx 0.8387$ - $\sin 74^\circ \approx 0.9613$ 6. **Substitute and compute:** $$x = \frac{22.4 \times 0.8387}{0.9613}$$ 7. **Simplify the fraction:** $$x = 22.4 \times \frac{0.8387}{0.9613}$$ 8. **Calculate the division:** $$\frac{0.8387}{0.9613} \approx 0.8729$$ 9. **Multiply:** $$x \approx 22.4 \times 0.8729 = 19.55$$ 10. **Round to the nearest tenth:** $$x \approx 19.6$$ 11. **Find the perimeter:** $$P = DE + EF + DF = 22.4 + 28 + 19.6 = 70.0$$ **Final answer:** The perimeter of $\triangle DEF$ is approximately $70.0$ units.