In a regular quadrilateral pyramid $S A B C D$, where each edge is equal to 2, a section is constructed by a plane parallel to the diagonal of the base $A C$ and the lateral edge $S B$ of the pyramid, intersecting the edge $A B$. Find the perimeter of the polygon obtained in this section, if the lower base of the section is equal to $\sqrt{2}$.
Let the secant plane intersect the diagonal $B D$ of the base of the pyramid at point $E$, and the lines $A B$ and $B C$ at points $M$ and $N$ respectively. Then the base plane passes through the line $A C$ parallel to the secant plane, and intersects the secant plane along the line $M N$, meaning $M N \| A C$. Moreover, according to the problem,

$$
M N=\sqrt{2}=\frac{1}{2} \cdot 2 \sqrt{2}=\frac{1}{2} A C
$$

Therefore, $M N$ is the midline of triangle $A B C$, i.e., $M$ and $N$ are the midpoints of edges $A B$ and $B C$ respectively. Let the secant plane intersect the lateral edge $S A$ at point $L$. Then the plane of the lateral face $A S B$ passes through the line $S B$, parallel to the secant plane, and intersects the secant plane along the line $M L$, meaning $M L \| S B$. Since $M$ is the midpoint of $A B$, $L$ is the midpoint of $S A$. Similarly, we can prove that the secant plane intersects the lateral edge $S C$ at its midpoint $K$. Therefore,

$$
M L=K N=\frac{1}{2} S B=1
$$

Let the secant plane intersect the line $S D$ at point $P$. Triangle $B S D$ is a right triangle because

$$
B D 2=(2 \sqrt{2}) 2=8=4+4=S B 2+S D 2
$$

Since $S D \perp M N$ and $P E \| S B$, the line $S B$ is perpendicular to the secant plane. Therefore, $L P \perp S D$, i.e., $L P$ is the perpendicular dropped from the midpoint $L$ of side $S A$ of the equilateral triangle $A S D$ with side length 2.

Thus, if $H$ is the midpoint of $S D$, then

$$
K P=L P=\frac{1}{2} A H=\frac{1}{2} \cdot \frac{2 \sqrt{3}}{2}=\frac{\sqrt{3}}{2}
$$

Therefore, the perimeter of the section is

$$
M N+M L+N K+L P+K P=\sqrt{2}+1+1+\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}=2+\sqrt{2}+\sqrt{3}
$$

## Answer

$2+\sqrt{2}+\sqrt{3}$.