Question

1. below is a composition of transformations to prove that \\( \triangle a b c \sim \triangle a ^ { \prime \prime } b ^ { \prime \prime } c ^ { \prime \prime } \\). the first transformation is a reflection over the \\( y \\)-axis. which rule shows the second transforma in this composition?

(a) \\( ( x, y ) \
ightarrow ( x + 2, y - 2 ) \\)
( \\( ( x, y ) \
ightarrow ( x + 6, y - 1 ) \\)
\\( ( x, y ) \
ightarrow \left( \frac { 1 } { 3 } x, \frac { 1 } { 3 } y \
ight) \\)
\\( ( x, y ) \
ightarrow ( 3 x, 3 y ) \\)

Explanation:

Step1: Analyze the transformation type

Since we are proving similarity ($\triangle ABC\sim\triangle A''B''C''$), and the first transformation is a reflection (a rigid - motion, which preserves shape and size). The second transformation should be a dilation (as dilation is a non - rigid transformation that changes the size but not the shape of a figure, which is required for similarity).

Step2: Check the scale factor

Let's assume a point. For example, if we consider the vertical distance (or horizontal distance) from the center of dilation (we can assume the origin for simplicity as it is a common center for dilations in coordinate - plane problems). Suppose the length of a side in $\triangle A'B'C'$ and $\triangle A''B''C''$. If we assume a side length in $\triangle A'B'C'$ (after reflection) and the corresponding side length in $\triangle A''B''C''$.
Let's take a general approach. If we have a point $(x,y)$ in $\triangle A'B'C'$ and the corresponding point $(x',y')$ in $\triangle A''B''C''$. A dilation rule is of the form $(x,y)\to(kx,ky)$ where $k$ is the scale factor.
Looking at the size of the triangles, $\triangle A''B''C''$ is smaller than $\triangle A'B'C'$. If we assume a vertex of $\triangle A'B'C'$ (say $B'$ which is at a higher position and larger in size compared to $B''$).
If we consider the coordinates (by visual inspection of the grid - like structure of the graph). Suppose a point $(x,y)$ in $\triangle A'B'C'$ and the corresponding point in $\triangle A''B''C''$. If we assume a point $(x,y)$ and its image $(x',y')$, and we know that for a dilation $(x',y')=(kx,ky)$. Since the image $\triangle A''B''C''$ is smaller, $k < 1$.
For option C: $(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$ is a dilation with a scale factor $k = \frac{1}{3}$.

Answer:

C. $(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$