Question

a sequence of transformations is performed on △xyz resulting in △xyz. the graphs are shown below. some transformations are listed below. • r90: a rotation 90° counter - clockwise • rx: a reflection across the x - axis • ry: a reflection across the y - axis which of these describes the sequence of transformations performed on △xyz that results in △xyz? b. r90 followed by ry c. ry followed by r90 d. r90 followed by rx

Explanation:

Step1: Analyze rotation effect

A $90^{\circ}$ counter - clockwise rotation ($R_{90}$) changes the coordinates of a point $(x,y)$ to $(-y,x)$.

Step2: Analyze x - axis reflection effect

A reflection across the x - axis ($R_x$) changes a point $(x,y)$ to $(x, - y)$.

Step3: Analyze y - axis reflection effect

A reflection across the y - axis ($R_y$) changes a point $(x,y)$ to $(-x,y)$.
Let's assume a general point on $\triangle XYZ$. If we first perform a $90^{\circ}$ counter - clockwise rotation ($R_{90}$) on a point $(x,y)$ getting $(-y,x)$ and then reflect across the x - axis ($R_x$), the new point is $(-y,-x)$.
If we first reflect across the x - axis ($R_x$) on a point $(x,y)$ getting $(x, - y)$ and then perform a $90^{\circ}$ counter - clockwise rotation ($R_{90}$), the new point is $(y,x)$.
If we first perform a $90^{\circ}$ counter - clockwise rotation ($R_{90}$) and then reflect across the y - axis ($R_y$), for a point $(x,y)$ after $R_{90}$ we have $(-y,x)$ and after $R_y$ we have $(y,x)$.
If we first reflect across the y - axis ($R_y$) on a point $(x,y)$ getting $(-x,y)$ and then perform a $90^{\circ}$ counter - clockwise rotation ($R_{90}$), the new point is $(-y,-x)$.

By observing the orientation and position of $\triangle XYZ$ and $\triangle X'Y'Z'$ on the coordinate - plane, we can see that a $90^{\circ}$ counter - clockwise rotation ($R_{90}$) followed by a reflection across the y - axis ($R_y$) will transform $\triangle XYZ$ to $\triangle X'Y'Z'$.

Answer:

A. $R_{90}$ followed by $R_y$