3. ( $7 \mathrm{p}$ ) In the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}, M$ is the midpoint of side $A B, C^{\prime} M \cap D^{\prime} B=\{P\}$. If $B P=4 \sqrt{3}$, find the distance from point $\mathrm{D}$ to the diagonal $D^{\prime} B$.
3. Desen

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Finding $C^{\prime}$ and $A \Rightarrow C^{\prime} A$ is the diagonal in the rectangle $A B C^{\prime} D^{\prime}, \text{with} \quad A C^{\prime} \cap B D^{\prime}=\{O\}$.

In triangle $A B C^{\prime}, C^{\prime} M \cap B O=\{P\}$

$P$ is the centroid $\Rightarrow B P=\frac{2}{3} B O \Rightarrow \frac{2}{3} \frac{l \sqrt{3}}{2}=B P \Rightarrow l=\frac{3 \cdot 4 \sqrt{3}}{\sqrt{3}}$

$\Rightarrow l=12 \quad$ (edge of the cube)

(2 p )

In triangle $D^{\prime} D B$ with $m(\angle D)=90^{\circ}$, we denote $\left(D, B D^{\prime}\right)=D E, D E=\frac{D D^{\prime} \cdot D B}{D^{\prime} B}=$ $4 \sqrt{6} \text{ cm}$