3. In a cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$ with edge length $a$, $X$ and $Y$ are the centers of the squares $A A_{1} B B_{1}$ and $B B_{1} C_{1} C$, respectively, and $Z$ is on the diagonal $B D$ such that $D Z=3 Z B$. Then the area of the section through $X$, $Y$, and $Z$ is $\qquad$
3. $\frac{\sqrt{2}}{2} a^{2}$.

According to the problem, the section through $X, Y, Z$ is a rectangle, with a length equal to the height of the cube $a$, and a width equal to the length of $XY$, which is $\frac{\sqrt{2}}{2} a$. Therefore, the area of the section through $X, Y, Z$ is $\frac{\sqrt{2}}{2} a^{2}$.