2. As shown in Figure 1, in the first quadrant, the three vertices $A, B, C$ of rectangle $A B C D$ are on the graphs of the functions $y=$ $\log _{\frac{\sqrt{2}}{2}} x$, $y=x^{\frac{1}{2}}$, and $y=\left(\frac{\sqrt{3}}{2}\right)^{x}$, respectively, and the sides of the rectangle are parallel to the coordinate axes. If the y-coordinate of point $A$ is 2, then the coordinates of point $D$ are
2. $\left(\frac{1}{2}, \frac{9}{16}\right)$.

Substitute $y_{A}=2$ into $y=\log _{\frac{\sqrt{2}}{2}} x$, we get $x_{A}=\frac{1}{2}$.
Thus, $x_{D}=\frac{1}{2}$.
Since $A B / / x$ axis, therefore, $y_{B}=y_{A}=2$.
Substitute into $y=x^{\frac{1}{2}}$, we get $x_{B}=4$. Thus, $x_{C}=4$.
Substitute $x_{c}=4$ into $y=\left(\frac{\sqrt{3}}{2}\right)^{x}$, we get $y_{c}=\frac{9}{16}$. Thus, $y_{D}=\frac{9}{16}$.
Therefore, point $D\left(\frac{1}{2}, \frac{9}{16}\right)$.