6. Calculate the area of the square $C D E F$ inscribed in the right triangle $A B C$ whose legs have lengths $|A C|=9 \mathrm{~cm}$ and $|B C|=6 \mathrm{~cm}$.
6. 

![](https://cdn.mathpix.com/cropped/2024_05_30_6b5fd03bf9bd38c6aa65g-2.jpg?height=443&width=828&top_left_y=1109&top_left_x=294)

From the given conditions and using the labels as shown in the figure, the area of the right triangle \( \triangle ABC \) is

\( P_{\triangle ABC} = \frac{1}{2} |AC| \cdot |BC| = \frac{1}{2} \cdot 9 \cdot 6 = 27 \, \text{cm}^2 \).

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Since the diagonal \( \overline{CE} \) of the given square divides the triangle \( \triangle ABC \) into two triangles ( \( \triangle CAE \) and \( \triangle BCE \) ), we have

\( P_{\triangle ABC} = P_{\triangle CAE} + P_{\triangle BCE} = \frac{1}{2} |AC| \cdot x + \frac{1}{2} |BC| \cdot x \).

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Thus, \( 27 = \frac{1}{2} \cdot 9 \cdot x + \frac{1}{2} \cdot 6 \cdot x \).

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Solving this equation, we find the side length of the square, which is \( x = 3.6 \, \text{cm} \).

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Finally, the area \( P \) of the square \( CDEF \) is \( P = x \cdot x = 3.6 \cdot 3.6 = 12.96 \, \text{cm}^2 \).

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TOTAL 10 POINTS