PROBLEM 3. Consider the regular quadrilateral pyramid $V A B C D$, with $A B=12$. Let $A C \cap B D=\{O\}$, $M$ be the midpoint of segment $[B C]$, and $P$ be the midpoint of segment $[A B]$. If the cosine of the dihedral angle formed by the planes $(V B C)$ and $(A B C)$ is equal to $\frac{3}{4}$, determine:

a) the distance from point $P$ to the plane $(V B C)$;

b) the distance from point $O$ to the plane (VPM);

c) the tangent of the angle formed by the planes $(V A C)$ and $(V B C)$.
PROBLEM 3. Consider the regular quadrilateral pyramid $VABCD$, with $AB=12$. Let $AC \cap BD=\{O\}$, $M$ be the midpoint of segment $[BC]$, and $P$ be the midpoint of segment $[AB]$. If the cosine of the dihedral angle formed by the planes $(VBC)$ and $(ABC)$ is equal to $\frac{3}{4}$, determine:

a) the distance from point $P$ to the plane $(VBC)$;

b) the distance from point $O$ to the plane $(VPM)$;

c) the tangent of the angle formed by the planes $(VAC)$ and $(VBC)$.

Grading rubric. Let $\alpha$ be the measure of the dihedral angle formed by the planes $(VBC)$ and $(ABC)$.

(1p) a) Since $\alpha=m(\widehat{VMO})$, and $\cos \alpha=\frac{3}{4}$, we obtain $VM=8$ and $VO=2\sqrt{7}$.

(2p) Since $PO \| BC$, it follows that $PO \|(VBC)$, hence $d(P; (VBC))=d(O; (VBC))=\frac{3\sqrt{7}}{2}$.

(1p) b) If $OB \cap PM=\{E\}$, then $OE=3\sqrt{2}$, $VE=\sqrt{46}$

(1p) and $d(O; (VPM))=d(O; VE)=\frac{6\sqrt{161}}{23}$.

(1p) c) If $S \in (VC)$, such that $OS \perp VC$, then the measure of the dihedral angle formed by the planes $(VAC)$ and $(VBC)$ is $\beta=m(\widehat{BSO})$.

(1p) From $OB=6\sqrt{2}$ and $OS=\frac{6\sqrt{14}}{5}$, we obtain $\operatorname{tg} \beta=\frac{OB}{OS}=\frac{5\sqrt{7}}{7}$.