B2. Given is a quadrilateral $A B C D$

with sides $A B=16, B C=21, C D=2$ and $D A=28$.

Furthermore, $A B$ is parallel to $C D$. Two lines that are parallel to $A B$ and $C D$ divide quadrilateral $A B C D$ into three similar quadrilaterals. Calculate the perimeter of the smallest of these three quadrilaterals.
B2. We denote the intersection points of the two lines with the quadrilateral as $P, Q, R$, and $S$, with $P$ and $R$ on $D A$ and $Q$ and $S$ on $C B$, see figure. The three quadrilaterals are similar. Therefore, we have:
$\frac{|D C|}{|P Q|}=\frac{|P Q|}{|R S|}=\frac{|R S|}{|A B|}$, so $|P Q|^{2}=2 \times|R S|, |R S|^{2}=|P Q| \times 16$ thus $|P Q|^{3}=64$ and $|P Q|=4$ and $|R S|=8$. The sides of quadrilateral $R S Q P$ are therefore twice as long as the corresponding sides of quadrilateral $P Q C D$, and the sides of quadrilateral $A B S R$ are again twice as long as the corresponding sides of quadrilateral $R S Q P$.

$|R P|=2 \times|P D|$ and $|A R|=2 \times|R P|=4 \times|P D|$. Thus, $|A D|=7 \times|P D|$, so $|P D|=4$. Similarly, you find: $|Q C|=3$.

The perimeter of quadrilateral $P Q C D$ is 13.

![](https://cdn.mathpix.com/cropped/2024_04_17_608b2a812c4856940d57g-2.jpg?height=511&width=394&top_left_y=193&top_left_x=1416)