10. (12 points) As shown in the figure, a rectangle with an area of 420 square centimeters is divided into five triangles by four line segments, and the areas of these five triangles $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ form an arithmetic sequence. Therefore, $S_{5}$ is $\qquad$ square centimeters.
【Solution】Solution: Let the areas of the five triangles $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ form an arithmetic sequence with a common difference of $d$, then $5 S_{5}-\frac{5 \times 4}{2} d=420$,

Rearranging, we get: $S_{5}-2 d=84$
According to the diagram, we have:
$$
S_{4}+S_{5}=2 S_{5}-d=420 \div 2=210 \text {, }
$$

So $2 S_{5}-d=210$
(2) $\times 2$ - (1), we get $3 S_{5}=336$,

Solving for $S_{5}=112$.
Therefore, $S_{5}$ is 112 square centimeters.
The answer is: 112.