$s$ is a reflection of the plane about $x=1$. $r$ is a reflection across the x-axis.

Question

$s$ is a reflection of the plane about the vertical line $x=1$. $r$ is a reflection across the x-axis. $g$ is an isometry and $grg^{-1}=s$. What is $g$?

Answer 1

Let $P=(a,b)$ a point. With respect to a coordinate system that has the two axis $s$ and $r$ as reference axis its coordinates are $P=(a-1,b)$ and $s(P)=(1-a,b)$.

Now, in this reference system, consider a rotation $R$ of $\pi/2$ with fixed point the common point of $s$ and $r$, we have: $$ R(P)=(-b,a-1) $$ $$ rR(P)=(-b,1-a) $$ $$ R^{-1}rR(P)=(1-a,b)=s(P) $$ so : $g= R^{-1}$: a rotation of $-\pi/2$ of center $(1,0)$.