Question

suppose p is the line with equation y = 4 and q is the line with equation y = 9. write the following composition of reflections as one translation.
rp ∘ rq
rp ∘ rq = t

Explanation:

Step1: Recall reflection - translation relationship

The composition of two reflections $R_p$ and $R_q$ over two parallel lines $p:y = 4$ and $q:y=9$ is a translation. The distance between the two parallel lines $d$ is given by the absolute - value of the difference of their $y$ - intercepts.

Step2: Calculate the distance between the lines

The distance $d=\vert9 - 4\vert=5$. When we have a composition of reflections $R_p\circ R_q$ over two parallel lines $p$ and $q$, the translation vector $\vec{v}$ has a magnitude of $2d$ in the direction perpendicular to the lines. Since the lines are horizontal, the translation is vertical.

Step3: Determine the translation vector

The distance between the two lines is $d = 5$, so the translation vector $\vec{v}$ for the composition $R_p\circ R_q$ is a vertical translation. The magnitude of the translation is $2d=10$ in the positive $y$ - direction (because the order of reflections matters). The translation $T$ can be written as $T(x,y)=(x,y + 10)$.

Answer:

A translation $T(x,y)=(x,y + 10)$