## SUBJECT III

Let $O$ and $O^{\prime}$ be the centers of the faces $A B C D$ and $B C C B^{\prime}$ of the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}, A B=a$.

a) Find the measure of the angle between the lines $D^{\prime} O$ and $A^{\prime} O^{\prime}$.

b) Calculate the distance from the point $O^{\prime}$ to the plane ($D^{\prime} A C$).

(prof. Bunu Iulian, Liceul de Arte Baia Mare)
## SUBJECT III

Let $O$ and $O^{\prime}$ be the centers of the faces $A B C D$ and $B C C B^{\prime}$ of the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}, A B=a$.

a) Find the measure of the angle between the lines $D^{\prime} O$ and $A^{\prime} O^{\prime}$.

b) Calculate the distance from the point $O^{\prime}$ to the plane $(D^{\prime} A C)$.

Scoring. a) Let $O^{\prime \prime}$ be the center of $A^{\prime} B^{\prime} C^{\prime} D$. We deduce $O^{\prime \prime} B / / D^{\prime} O$ 0.5 p

Let $M$ be the midpoint of $O^{\prime} C^{\prime}$. We deduce $O^{\prime \prime} M / / A^{\prime} O^{\prime}$. 0.5 p

The measure of the angle between the lines $D^{\prime} O$ and $A^{\prime} O^{\prime}$ is $m\left(\angle B O^{\prime \prime} M\right)$. 0.5 p

Triangle $A^{\prime} B C$ is equilateral $\Rightarrow A^{\prime} O^{\prime} \perp B C$ 0.5 p

$\Rightarrow O^{\prime \prime} M / / A^{\prime} O^{\prime}$ and $O^{\prime \prime} M \perp B C$ 1 p

$\Rightarrow\left(\angle B O^{\prime} M\right)=60^{\circ}$ 1 p

b) If $p r_{\left(D^{\prime} A C\right)} B^{\prime}=E, p r_{\left(D^{\prime} A C\right)} O^{\prime}=F \Rightarrow O^{\prime} F=\frac{1}{2} B^{\prime} E$ 1 p

Find $B^{\prime} E=\frac{2 \sqrt{3} a}{3}$

1.5 p

Finalization: $O^{\prime} F=d\left(O^{\prime} ;\left(D^{\prime} A C\right)\right)=\frac{\sqrt{3} a}{3}$.