4. Consider the isosceles trapezoid $A B C D$ with $A B \parallel C D, A C \cap B D=\{O\}, A B=40 \mathrm{~cm}$, $C D=\frac{1}{4} A B$ and the diagonals are perpendicular. On the plane of the trapezoid, the perpendicular $O M$ is raised such that $O M=5 \sqrt{2} \mathrm{~cm}$.

(3p) a) Calculate the area of triangle $M C D$.

(4p) b) Determine $\sin (\angle(B M, A D))$.

Note: All questions are mandatory.

Time allowed for work: 3 hours.
4. a) $O Q \perp C D$; according to $\mathrm{T} 3 \perp \Rightarrow M Q \perp C D$

(1p)

We calculate $C D=10 \mathrm{~cm}, O Q=5 \mathrm{~cm}$, and in $\triangle M O Q$ right at $O \Rightarrow M Q=5 \sqrt{3} \mathrm{~cm} \ldots \ldots$. (1p)

$$
A_{M C D}=\frac{C D \cdot M Q}{2}=25 \sqrt{3} \mathrm{~cm}^{2}
$$

(1p)

b) $B S \square A D, S \in C D \Rightarrow \square(B M, A D)=\square M B S$

(1p) $A O=B O=20 \sqrt{2} \mathrm{~cm}, D O=5 \sqrt{2} \mathrm{~cm}, A D=B S=5 \sqrt{34} \mathrm{~cm}, O S=25 \sqrt{2} \mathrm{~cm}$

(1p)

In $\triangle M O B$ right at $O \Rightarrow M B=5 \sqrt{34} \mathrm{~cm}$, and in $\triangle M O S$ right at $O \Rightarrow M S=10 \sqrt{13} \mathrm{~cm}$...

(1p)

In $\triangle B M S$ isosceles $d(B, M S)=5 \sqrt{21} \mathrm{~cm}$, then from $A_{B M S}=\frac{M S \cdot d(B, M S)}{2}=$ $\frac{B M \cdot B S \cdot \sin (\square M B S)}{2} \Rightarrow \sin (\square M B S)=\frac{\sqrt{273}}{17}$

(1p)