Question

which rule describes the composition of transformations that maps △def to △def?
$r_{0,90^{circ}}circ t_{5,0}(x,y)$
$t_{ - 5,0}circ r_{0,90^{circ}}(x,y)$
$t_{5,0}circ r_{0,90^{circ}}(x,y)$
$r_{0,90^{circ}}(x,y)circ t_{ - 5,0}$

Explanation:

Step1: Analyze the rotation

First, observe that from $\triangle DEF$ to $\triangle D'E'F'$, the triangle is rotated $90^{\circ}$ counter - clockwise about the origin. The rule for a $90^{\circ}$ counter - clockwise rotation about the origin $R_{0,90^{\circ}}(x,y)$ is $(x,y)\to(-y,x)$.

Step2: Analyze the translation

Then, from $\triangle D'E'F'$ to $\triangle D''E''F''$, the triangle is translated 5 units to the left. The rule for a translation $T_{- 5,0}(x,y)$ is $(x,y)\to(x - 5,y)$.

Step3: Determine the composition of transformations

The composition of transformations should first perform the rotation and then the translation. So the rule is $T_{-5,0}\circ R_{0,90^{\circ}}(x,y)$.

Answer:

$T_{-5,0}\circ R_{0,90^{\circ}}(x,y)$