Question

compositions of reflections
review
find a composition of two reflections that maps △xyz onto △xyz.
△xyz is a translation of △xyz, so it is a composition of two reflections.
the line containing x and x is horizontal, so the reflection lines must be perpendicular to that line, or
if you reflect △xyz across the line x = -2, the image is the dashed triangle shown.
the point x is units from its image x. the distance between the reflection lines is half that, or 3.5 units. the second reflection line is x = 1.5.
if you reflect the dashed triangle over the line x = , the image is △xyz.
the composition is (r_{x = 1.5} ∘ r_{x=-2})(△xyz)=△xyz.

Explanation:

Step1: Determine the orientation of reflection lines

Since the line containing $X$ and $X'$ is horizontal, reflection lines must be vertical.

Step2: Calculate the distance between $X$ and $X'$

The distance between the reflection - lines is 3.5 units and it is half of the distance between $X$ and $X'$. So the distance between $X$ and $X'$ is $3.5\times2 = 7$ units.

Step3: Identify the second reflection line

We know the first reflection line is $x=-2$ and the distance between the two reflection lines is 3.5 units. The second reflection line is $x=-2 + 3.5=1.5$. When we reflect the dashed - triangle (image after first reflection) over the line $x = 1.5$, we get $\triangle X'Y'Z'$.

Answer:

The first blank: vertical; the second blank: 7; the third blank: 1.5