1. Given point $A(2,1)$, determine point $P$ on the coordinate axes such that $\triangle A O P$ is an isosceles triangle. Then the number of points that satisfy this condition is $(\quad)$.
(A) 4
(B) 6
(C) 8
(D) More than 8
$-、 1 . C$.
(1) With $A$ as the center and $O A$ as the radius, draw $\odot A$ intersecting the $x$-axis and $y$-axis at $P_{1}$ and $P_{2}$, then we have $P_{1}(4,0), P_{2}(0,2)$.
(2) With $O$ as the center and $O A$ as the radius, draw $\odot O$ intersecting the $x$-axis and $y$-axis at $P_{3}$, $P_{4}$, $P_{5}$, and $P_{6}$, then we have $P_{3}(\sqrt{5}, 0)$, $P_{4}(-\sqrt{5}, 0)$, $P_{5}(0, \sqrt{5})$, $P_{6}(0,-\sqrt{5})$.
(3) Draw the perpendicular bisector of $O A$ intersecting the $x$-axis and $y$-axis at $P_{7}$ and $P_{8}$, then we have $P_{7}\left(\frac{5}{4}, 0\right)$, $P_{8}\left(0, \frac{5}{2}\right)$.