Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$?
I notice that the reflection changed the $y$-coordinates of the vertices, but not the $x$-coordinates. This suggests that line $L$ is horizontal and has the form $y = k$ for some constant $k$. To find $k$, I need to find the midpoint of the segment connecting each vertex and its image. For example, the midpoint of $AA'$ is $(2, -1)$, since $(2 + 2)/2 = 2$ and $(3 + (-5))/2 = -1$. Similarly, the midpoint of $BB'$ is $(7, -1)$ and the midpoint of $CC'$ is $(-4, -1)$. This means that all three midpoints lie on line $L$, and the value of $k$ is $-1$. Therefore, the equation of line $L$ is $y = -1$. 