10.068. The length of the base of the triangle is 36 cm. A line parallel to the base divides the area of the triangle in half. Find the length of the segment of this line enclosed between the sides of the triangle.
## Solution.

Let the required length be $x$. Then, from the similarity of triangles $A B C$ and $L B M$ (Fig. 10.67), we have $A C^{2}: x^{2}=S:(S / 2)$, where $S$ is the area of $\triangle A B C$. Hence, $A C^{2}=2 x^{2}$ and $x=18 \sqrt{2}$ (cm).

Answer: $18 \sqrt{2}$ cm.