21. As shown in Figure 2, fold the square $A B C D$ with side length 1. Vertex $C$ lands on point $C^{\prime}$ on side $A D$, and $B C$ intersects $A B$ at point $E$. If $C^{\prime} D$ $=\frac{1}{3}$, then the perimeter of $\triangle A E C^{\prime}$ is ( ).
(A) 2
(B) $1+\frac{2 \sqrt{3}}{3}$
(C) $\frac{7}{3}$
(D) $\sqrt{136}$
(E) $1+\frac{3 \sqrt{3}}{4}$
21. A.

Let $N$ be the intersection of the crease with $CD$, and let $DN=x$. Then $C'N = CN = 1 - x$.
In $\triangle C' DN$,
$$
\left(\frac{1}{3}\right)^{2} + x^{2} = (1 - x)^{2} \Rightarrow x = \frac{4}{9}.
$$

By $\triangle AEC' \sim \triangle DC'N$,
$$
\Rightarrow \frac{AC'}{DN} = \frac{AE}{C'D} = \frac{C'E}{C'N}.
$$

Given $AC' = \frac{2}{3}, C'N = \frac{5}{9}$, we have
$$
AE = \frac{1}{2}, C'E = \frac{5}{6}.
$$

Thus, $AE + AC' + C'E = \frac{3+4+5}{6} = 2$.