8,9
[ Rectangles and Squares. Properties and Characteristics]
The side of the square is 1. A line is drawn through its center. Calculate the sum of the squares of the distances from
the four vertices of the square to this line.
Let a line passing through the center $O$ of the square $A B C D$ intersect the side $A B$. Drop perpendiculars $A P$ and $B Q$ to this line. Triangles $A P O$ and $O Q B$ are equal by hypotenuse and acute angle. Therefore, $A P^{2}+B Q^{2}=$ $A P^{2}+O P^{2}=A O^{2}=1 / 2$

## Answer

1.