9) A triangle has two vertices at the points with coordinates $(-4,1)$ and $(2,-1)$, and the third vertex at the point with coordinates $(1, k)$. For how many real values of $k$ is this triangle isosceles?
(A) None
(B) 1
(C) 4
(D) 5
(E) Infinite.
9) The answer is $(\mathbf{D})$. Let $P=(-4,1), Q=(2,-1), r$ be the line $x=1$. Intersecting $r$ with the perpendicular bisector of $P Q$ yields a point $R_{1}$ such that the triangle $P Q R_{1}$ is isosceles with base $P Q$.

The circle with center $P$ and radius $P Q$ intersects the line $r$ at two points $R_{2}$ and $R_{3}$ such that each of the triangles $P Q R_{2}, P Q R_{3}$ is isosceles, with the sides meeting at $P$ being equal to each other. The circle with center $Q$ and radius $P Q$ intersects the line $r$ at two points $R_{4}$ and $R_{5}$ such that each of the triangles $P Q R_{4}, P Q R_{5}$ is isosceles, with the sides meeting at $Q$ being equal to each other. Therefore, there are five isosceles triangles of the desired type.