1. In Rt $\triangle A B C$, it is known that $\overrightarrow{A B}=(2,3), \overrightarrow{A C}$ $=(1, k)$. Then the value of $k$ is
$-1, -\frac{2}{3}, \frac{11}{3}, \frac{3 \pm \sqrt{13}}{2}$.
Discuss in three cases.
(1) When $\angle A=\frac{\pi}{2}$, from $\overrightarrow{A B} \cdot \overrightarrow{A C}=0$, we get
$$
2 \times 1+3 k=0 \Rightarrow k=-\frac{2}{3} \text {. }
$$
(2) When $\angle B=\frac{\pi}{2}$, from the given, we have $\overrightarrow{B C}=\overrightarrow{A C}-$ $\overrightarrow{A B}=(-1, k-3)$, and by $\overrightarrow{A B} \cdot \overrightarrow{B C}=0$, we get
$$
2 \times(-1)+3(k-3)=0 \Rightarrow k=\frac{11}{3} \text {. }
$$
(3) When $\angle C=\frac{\pi}{2}$, from $\overrightarrow{A C} \cdot \overrightarrow{B C}=0$, we get
$$
-1+k(k-3)=0 \Rightarrow k=\frac{3 \pm \sqrt{13}}{2} \text {. }
$$