Question

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

Explanation:

Step1: Analyze rotation

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

Step2: Analyze 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 5 units to the left is $T_{ - 5,0}(x,y)=(x - 5,y)$.

Step3: Determine the composition

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

Answer:

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