## Subject 3.

Let $\mathrm{ABCD}$ be a rhombus with $A B=6 \mathrm{~cm}$ and $m(\hat{A})=45^{\circ}$. At point $\mathrm{O}$, the center of the circumcircle of triangle $\mathrm{ABD}$, the perpendicular $\mathrm{OM}$ is raised to the plane of the rhombus, $O M=4 \mathrm{~cm}$. Find:

a) The distance from point O to the plane (MBC).

b) The cosine of the angle between the planes (MBC) and (ABC).

a) Let $d$ be the perpendicular bisector of side (AD), and $\{O\}=d \cap A C$ the center of the circumcircle

of triangle $\mathrm{ABD},\{N\}=d \cap A D,\{P\}=d \cap B C,\{R\}=d \cap A B$.

Geometric figure.

Justification $M P \perp B C, d(O,(M B C))=O Q$

$\triangle A N R$ is a right-angled and isosceles triangle, $A N=N R=3, A R=3 \sqrt{2} ; \triangle B P R$ is a right-angled and isosceles triangle, $R B=A B-A R=3(2-\sqrt{2}), R P=P B=3(\sqrt{2}-1)$

$\triangle A O N \sim \triangle C O P \Rightarrow \frac{N O}{O P}=\frac{A N}{C P} \Leftrightarrow \frac{N O}{O P}=\frac{1}{1+\sqrt{2}} \Leftrightarrow O P=3 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

$\triangle M O P$ is a right-angled triangle, $M P=5, O Q=\frac{O M \cdot O P}{M P}=\frac{12}{5}$

$\mathbf{1 p}$

b) The angle between the planes (MBC) and (ABC) is $\angle O P M$ (justification)

$$
\cos (\angle O P M)=\frac{3}{5}
$$
3. Let $ABCD$ be a rhombus with $AB=6 \, \text{cm}$ and $m(\hat{A})=45^{\circ}$. At point $O$, the center of the circumcircle of triangle $ABD$, the perpendicular $OM$ is raised to the plane of the rhombus, $OM=4 \, \text{cm}$. Find:

a) The distance from point $O$ to the plane $(MBC)$.

b) The cosine of the angle between the planes $(MBC)$ and $(ABC)$.

Prof. Marinela Mihali