3. Let the rectangle $A B C D$ with $A D=4 \text{ cm}, A B=4 \sqrt{3} \text{ cm}$ and $m(A \hat{B} D)=30^{\circ}$. We denote by $M$ and $N$, respectively by $P$ and $Q$, the feet of the perpendiculars from $A$ and $C$ to the diagonal $B D$, respectively from $B$ and $D$ to the diagonal $A C$.

(4p) a) Show that $MQNP$ is a rectangle.

(3p) b) Calculate the area of the rectangle $M Q N P$.

Mihaela Diana Dragoie
3. a) Correct figure

![](https://cdn.mathpix.com/cropped/2024_06_07_2fe05cd0e210b2a6d5aeg-2.jpg?height=440&width=699&top_left_y=1576&top_left_x=290)

$\triangle A M D \equiv \triangle C N B \equiv \triangle B P C \equiv \triangle D Q A \Rightarrow D M=B N=C P=A Q$

$A O=B O=C O=D O \Rightarrow M O=N O=P O=Q O$ (differences of congruent segments) $\Rightarrow M Q N P$ rectangle

b) $\triangle A D O$ is isosceles with an angle of $60^{\circ}$, so it is equilateral, the altitudes $A M$ and $D Q$ are medians, $M Q$ is the midline $\Rightarrow M Q=\frac{D A}{2}$

In $\triangle A O B \quad N_{\text {_ is the midline } \Rightarrow Q N}^{-}=\frac{A B}{2}$

$\mathrm{A}_{M Q N P}=M Q \cdot Q N=2 \cdot 2 \sqrt{3}=4 \sqrt{3} \mathrm{~cm}^{2}$