Example 21. Divide a line segment of length $10 \mathrm{~cm}$ into two parts, use one part as the side length of an equilateral triangle to construct an equilateral triangle, and use the other part as the side length of a square to construct a square. Find the minimum value of the sum of their areas.

Divide a line segment of length $10 \mathrm{~cm}$ into two parts, use one part as the side length of an equilateral triangle to construct an equilateral triangle, and use the other part as the side length of a square to construct a square. Find the minimum value of the sum of their areas.
$(10-\mathrm{x}) \mathrm{cm}$. For the
$\mathrm{S}_{\text {equilateral triangle }}=\frac{1}{2} \cdot \frac{\sqrt{3}}{2}x^{2}=\frac{\sqrt{3}}{4} x^{2}$;
$\mathrm{S}_{\text {square }}=(10-\mathrm{x})^{2}$.
Let their combined area be $\mathrm{sm}^{2}$, then
$$
\begin{array}{l}
\mathrm{S}=\frac{\sqrt{3}}{4} \mathrm{x}^{2}+(10-\mathrm{x})^{2}=\left(\frac{\sqrt{3}}{4}+1\right) \mathrm{x}^{2} \\
-20 \mathrm{x}+100 . \\
\because \frac{\sqrt{3}}{4}+1>0 \quad \therefore \text { S has a minimum value. } \\
\mathrm{S}_{\text {minimum value }}=\frac{4 \mathrm{ac}-\mathrm{b}^{2}}{4 \mathrm{a}}=\frac{100(4 \sqrt{3}-3)}{13}\left(\mathrm{~cm}^{2}\right) .
\end{array}
$$