10. (10 points) In triangle $ABC$, $AB=180$ cm, $AC=204$ cm, $D, F$ are points on $AB$, $E, G$ are points on $AC$, connecting $CD, DE, EF, FG$, divides triangle $ABC$ into five smaller triangles of equal area. What is the length of $AF + AG$ in centimeters?
【Analysis】Given a fixed height, the ratio of the corresponding bases equals the ratio of the areas. Since the areas of the five small triangles are equal, $S_{\triangle A D C} = 4 S_{\triangle D B C}$, so $A D = 4 B D = 4 \times (180 \div 5) = 144$ (cm); Similarly, we can find the lengths of $A E$, $A F$, and $A G$, and then calculate the length of $A F + A G$.

【Solution】In $\triangle A B C$, since $S_{\triangle A D C} = 4 S_{\triangle D B C}$, we have $A D = 4 B D = 4 \times (180 \div 5) = 144$ (cm);

In $\triangle A D C$, since $S_{\triangle A D E} = 3 S_{\triangle E D C}$, we have $A E = 3 E C = 3 \times (204 \div 4) = 153$ (cm); In $\triangle A D E$, since $S_{\triangle A F E} = 2 S_{\triangle E F D}$, we have $A F = 2 D F = 2 \times (144 \div 3) = 96$ (cm); In $\triangle A F E$, since $S_{\triangle A F G} = S_{\triangle G F E}$, we have $A G = G E = 153 \div 2 = 76.5$ (cm); Therefore, $A F + A G = 96 + 76.5 = 172.5$ (cm);

Answer: $A F + A G$ is 172.5 cm.