1. **State the problem:** We are given two triangles sharing vertex $G$, with $HI \parallel EF$. Given lengths are $EF=4$, $HI=3$, $GF=5.6$, and $EG=4.8$. We need to find the length of $HG$. 2. **Use the properties of parallel lines and similar triangles:** Since $HI \parallel EF$, triangles $HGI$ and $EGF$ are similar by the AA criterion. 3. **Set up the ratio of corresponding sides:** The ratio of the sides $HI$ to $EF$ is $$\frac{HI}{EF} = \frac{3}{4}.$$ Since the triangles are similar, the ratio of corresponding segments from $G$ to $H$ and $G$ to $E$ must be the same as the ratio of $HI$ to $EF$: $$\frac{HG}{EG} = \frac{HI}{EF} = \frac{3}{4}.$$ 4. **Solve for $HG$:** $$HG = EG \times \frac{3}{4} = 4.8 \times \frac{3}{4}.$$ 5. **Calculate:** $$HG = 4.8 \times 0.75 = 3.6.$$ **Final answer:** $$\boxed{3.6}.$$