Question

how far apart are two parallel lines q and r such that $t_{(10,0)}(\triangle opq)=(r_{r}circ r_{q})(\triangle opq)$? the lines q and r are $square$ unit(s) apart. (type an integer or a decimal.)

Explanation:

Step1: Analyze the transformation notation

The notation $T_{(10,0)}$ represents a translation by the vector $(10,0)$. If $T_{(10,0)}(\triangle OPQ)=(r_{1}\circ r_{2})(\triangle OPQ)$, and we are dealing with parallel - lines $q$ and $r$. A composition of two reflections $r_{1}\circ r_{2}$ over two parallel lines is equivalent to a translation. The distance between two parallel lines $q$ and $r$ such that a composition of reflections over them is equivalent to a translation $T_{(d,0)}$ (where $d$ is the magnitude of the translation vector in the direction perpendicular to the lines) is given by half of the magnitude of the translation vector when the lines are parallel.

Step2: Calculate the distance between the lines

The translation vector is $(10,0)$. The distance $d_{lines}$ between the two parallel lines $q$ and $r$ is $\frac{|10|}{2}=5$.

Answer:

5