4. Consider $A B C D$ to be a rhombus with diagonals $A C=8 \text{ cm}, B D=6 \text{ cm}$. At point $N$, the midpoint of segment $A O, A C \cap B D=O$, the perpendicular $M N$ is raised to the plane of the rhombus, $M N=2 \sqrt{3} \text{ cm}$. Calculate:

(3p) a) The area of triangle $M B D$.

(4p) b) The distance from point $M$ to side $A D$.

Note: All questions are mandatory.

Actual working time: 3 hours.

## Grading Rubric OLM Grade 8, 2014
4. ..... a) $\triangle M N A \ll N=90^{\circ} \Rightarrow M A=4 \mathrm{~cm}$
.(1p)
$\triangle M O A$ equilateral $\Rightarrow M O=4 \mathrm{~cm} \Rightarrow A_{\triangle M B D}=12 \mathrm{~cm}^{2}$ ..... (2p)
b) $M N \perp(A O D), N T \perp A D \Rightarrow M T \perp A D \Rightarrow d(M, A D)=M T$. ..... (1p)

$$
O R \text { altitude in } \triangle A O D, O R=\frac{12}{5} \mathrm{~cm}
$$

$N T$ midline in $\triangle A O R \Rightarrow N T=\frac{6}{5} \mathrm{~cm}$

$\triangle M N T: M T=\frac{4 \sqrt{21}}{5} \mathrm{~cm}$

(1p )