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 translation

First, observe that from $\triangle DEF$ to $\triangle D'E'F'$, the triangle is translated 5 units to the right. The rule for a translation 5 units to the right is $T_{5,0}(x,y)=(x + 5,y)$.

Step2: Analyze rotation

Then, from $\triangle D'E'F'$ to $\triangle D''E''F''$, the triangle is rotated 90 - degrees counter - clockwise about the origin. The rule for a 90 - degree counter - clockwise rotation about the origin is $R_{0,90^{\circ}}(x,y)=(-y,x)$. In a composition of transformations, the transformation that is done first is on the right - hand side of the composition symbol $\circ$. So the composition of transformations that maps $\triangle DEF$ to $\triangle D''E''F''$ is $R_{0,90^{\circ}}\circ T_{5,0}(x,y)$.

Answer:

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