1. **State the problem:** We have triangle EGD with point F on segment EG such that EF = $x$ and FG = $x + 10$. We know ED = 24 and GD = 54. We want to find the length EF. 2. **Identify the approach:** Since F lies on EG, and we know lengths ED and GD, we can use the Angle Bisector Theorem if angle D bisects segment EG at F. The theorem states: $$\frac{EF}{FG} = \frac{ED}{GD}$$ 3. **Apply the theorem:** Substitute the known values: $$\frac{x}{x + 10} = \frac{24}{54}$$ 4. **Simplify the ratio on the right:** $$\frac{24}{54} = \frac{4}{9}$$ 5. **Set up the equation:** $$\frac{x}{x + 10} = \frac{4}{9}$$ 6. **Cross multiply:** $$9x = 4(x + 10)$$ 7. **Distribute the right side:** $$9x = 4x + 40$$ 8. **Subtract $4x$ from both sides:** $$9x - \cancel{4x} = \cancel{4x} + 40 - 4x$$ $$5x = 40$$ 9. **Divide both sides by 5:** $$\frac{5x}{\cancel{5}} = \frac{40}{\cancel{5}}$$ $$x = 8$$ 10. **Conclusion:** The length EF is $8$ units.