Subject 3. In the equilateral triangle $\mathrm{ABC}$ with $\mathrm{AB}=6 \mathrm{~cm}$, perpendiculars $\mathrm{A}^{\prime} \mathrm{A} \perp(\mathrm{ABC}), \mathrm{B}^{\prime} \mathrm{B} \perp(\mathrm{ABC})$ are raised from the same side of the plane (ABC), such that $\mathrm{AA}^{\prime}=\mathrm{BB}^{\prime}=6 \sqrt{3} \mathrm{~cm}$.

a) Find the distance from A to the plane (A'BC).

b) Find the sine of the angle between A'B and B'C.
## Subject 3.

| a. | Construction of the perpendicular from A to (A'BC) |  |
| :--- | :--- | :---: |
|  | Calculation of the distance $: \frac{6 \sqrt{15}}{5}$ | **3 p** |
| b. | Extend $[\mathrm{AB}]$ with $[\mathrm{BE}]$, $\mathrm{AB}=\mathrm{BE} \Rightarrow \mathrm{A}^{\prime} \mathrm{B} \\| \mathrm{B}^{\prime} \mathrm{E} \Rightarrow \varangle\left(A^{\prime} B, B^{\prime} C\right)=\varangle C B^{\prime} E$ |  |
|  | Area of triangle $\mathrm{B}^{\prime} \mathrm{CE}$ in two ways and finalizing $\sin \left(E B^{\prime} C\right)=\frac{\sqrt{39}}{8}$ | **4 p** |