2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).
(A) $\frac{\sqrt{5}+1}{2}$
(B) $\frac{\sqrt{5}-1}{2}$
(C) 1
(D) 2
2. A. Since $\triangle B O C \sim \triangle A B C$, we have $\frac{B O}{A B}=\frac{B C}{A C}$, which means
$$
\frac{1}{a}=\frac{a}{a+1} \Rightarrow a^{2}-a-1=0 .
$$

Given $a>0$, solving yields $a=\frac{1+\sqrt{5}}{2}$.