2. In the tetrahedron $A B C D$ with all edges congruent, let $D O \perp(A B C), O \in(A B C)$. The point $M$ is the projection of point $O$ onto the edge $[D B]$, and $M C=2 \sqrt{7} \mathrm{~cm}$. Calculate the value of the sine of the angle between the line $M C$ and the plane (BOD).

Testarea Naţională, 2007
## Solution:

Let $A B=l$.

$B O=\frac{l \sqrt{3}}{3}$ and from the leg theorem in triangle $D O B$ we get $B M=\frac{l}{3}$

Let $C S \perp B D, S \in(B D)$. We obtain $C S=\frac{l \sqrt{3}}{2}$ and $S M=\frac{l}{2}-\frac{l}{3}=\frac{l}{6}$

Using the Pythagorean theorem in triangle $S M C$, we find $l=6$

$1 \mathbf{p}$

$B O \cap A C=\{T\} \Rightarrow m(\Varangle C M ;(B O D))=m(\Varangle C M T)$

$2 p$

In the right triangle $C M T, \sin (\Varangle C M T)=\frac{T C}{C M}=\frac{3}{2 \sqrt{7}}=\frac{3 \sqrt{7}}{14}$

$1 p$