1. **State the problem:** We need to calculate the length of segment $FG$ in the given geometric figure. 2. **Analyze the figure:** We have a right triangle $DEF$ with right angle at $F$. The sides are $DF=3$ cm, $FE=4$ cm, and $GE=12$ cm with point $G$ on line $DE$. 3. **Calculate the length of $DE$:** Since $DEF$ is a right triangle with legs $DF=3$ cm and $FE=4$ cm, use the Pythagorean theorem: $$DE=\sqrt{DF^2 + FE^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}$$ 4. **Locate point $G$ on $DE$:** Given $GE=12$ cm, but $DE=5$ cm, so $G$ lies beyond $E$ on the line $DE$ extended. 5. **Find $DG$:** Since $G$ is on the line through $D$ and $E$, and $GE=12$ cm, then $$DG = DE + GE = 5 + 12 = 17 \text{ cm}$$ 6. **Use similar triangles:** The problem mentions right angles forming two smaller right triangles within $DEF$. The smaller triangle with hypotenuse $FG$ is similar to $DEF$. 7. **Calculate $FG$ using similarity:** Since $DF=3$ cm corresponds to $FG$, and $DG=17$ cm corresponds to $DE=5$ cm, the scale factor is $$\frac{DG}{DE} = \frac{17}{5}$$ Therefore, $$FG = DF \times \frac{DG}{DE} = 3 \times \frac{17}{5} = \frac{51}{5} = 10.2 \text{ cm}$$ **Final answer:** $$FG = 10.2 \text{ cm}$$