Question

suppose k is the line with equation x = - 8 and t is the line with equation x = - 5. write the following composition of reflections as one translation. r_t ∘ r_k r_k ∘ r_t = t □□

Explanation:

Step1: Recall reflection - translation relationship

The composition of two reflections $R_l\circ R_k$ over two parallel lines $k:x = a$ and $l:x = b$ is a translation. If the lines $k:x=-8$ and $l:x = - 5$ are parallel, and the distance between two parallel lines $x = a$ and $x = b$ is $d=\vert b - a\vert$.

Step2: Calculate the distance between the lines

The distance between the lines $x=-8$ and $x=-5$ is $d=\vert-5-(-8)\vert=\vert-5 + 8\vert=3$. The composition of reflections $R_l\circ R_k$ over two parallel lines $k$ and $l$ in the $x -$ direction is a translation $T$ in the $x -$ direction. The direction of the translation is from $k$ to $l$ (since we first reflect over $k$ and then over $l$).

Step3: Determine the translation vector

For two parallel vertical lines $x = a$ and $x = b$ with $b>a$, the composition $R_l\circ R_k$ is a translation $T$ with a vector $\vec{v}=(2(b - a),0)$. Here, $a=-8$ and $b = - 5$, so $2(b - a)=2(-5+8)=6$. The translation $T$ has a vector $\vec{v}=(6,0)$.

Answer:

A translation $T$ with vector $(6,0)$