1. **State the problem:** We have a circle with center F, points E and G on the circumference, and the angle $\angle EFG = 54^\circ$. Given $EF = 13$ units, find the length of $EG$.\n\n2. **Understand the setup:** Since E and G lie on the circle and F is the center, $EF$ and $FG$ are radii of the circle. Therefore, $EF = FG = 13$ units.\n\n3. **Identify the triangle:** Triangle $EFG$ is isosceles with sides $EF = FG = 13$ and angle between them $54^\circ$. We want to find the length $EG$, the chord opposite the angle.\n\n4. **Use the Law of Cosines:** For triangle $EFG$,\n$$EG^2 = EF^2 + FG^2 - 2 \cdot EF \cdot FG \cdot \cos(\angle EFG)$$\nSubstitute values:\n$$EG^2 = 13^2 + 13^2 - 2 \cdot 13 \cdot 13 \cdot \cos(54^\circ)$$\n\n5. **Calculate:**\n$$EG^2 = 169 + 169 - 338 \cdot \cos(54^\circ) = 338 - 338 \cdot \cos(54^\circ)$$\n\n6. **Evaluate $\cos(54^\circ)$:**\nUsing a calculator, $\cos(54^\circ) \approx 0.5878$.\n\n7. **Substitute and simplify:**\n$$EG^2 = 338 - 338 \times 0.5878 = 338 - 198.68 = 139.32$$\n\n8. **Find $EG$:**\n$$EG = \sqrt{139.32} \approx 11.80$$\n\n**Final answer:** The length of $EG$ is approximately **11.80** units.