Question

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

Explanation:

Step1: Analyze rotation

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

Step2: Analyze translation

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

Step3: Determine composition order

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

Answer:

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