Question

a sequence of transformations is performed on △def, resulting in △def. both triangles are shown on the coordinate plane below. some transformations are listed below. • ry: a reflection across the y - axis • rx: a reflection across the x - axis • t-6,0: a translation so that (x, y)→(x - 6, y). which of these describes the sequence of transformations performed on △def that results in △def? a. ry followed by t-6,0 b. rx followed by ry c. t-6,0 followed by rx d. t-6,0 followed by ry

Explanation:

Step1: Analyze reflection across y - axis

A reflection across the y - axis changes the sign of the x - coordinate of each point, i.e., $(x,y)\to(-x,y)$.

Step2: Analyze translation $T_{- 6,0}$

The translation $T_{-6,0}$ moves each point 6 units to the left, so $(x,y)\to(x - 6,y)$.

Step3: Check option A

If we first perform $R_y$: $(x,y)\to(-x,y)$, and then $T_{-6,0}$: $(-x,y)\to(-x - 6,y)$. This can match the transformation from $\triangle DEF$ to $\triangle D'E'F'$.

Step4: Check option B

$R_x$: $(x,y)\to(x,-y)$ followed by $R_y$: $(x,-y)\to(-x,-y)$. This is a rotation about the origin, not the correct transformation.

Step5: Check option C

$T_{-6,0}$: $(x,y)\to(x - 6,y)$ followed by $R_x$: $(x - 6,y)\to(x - 6,-y)$. This is not the correct transformation.

Step6: Check option D

$T_{-6,0}$: $(x,y)\to(x - 6,y)$ followed by $R_y$: $(x - 6,y)\to(-(x - 6),y)=(6 - x,y)$. This is not the correct transformation.

Answer:

A. $R_y$ followed by $T_{-6,0}$