Question

what composition of rigid motions maps △pqr to △xyz?
t<1,3> ∘ r(270°,o)
ry = -2 ∘ t<6,0>
t<6,2> ∘ rx = -2
rx = 0 ∘ t<0,2>

Explanation:

Step1: Analyze translation and reflection

Translation $T_{}$ moves a figure $a$ units horizontally and $b$ units vertically. Reflection $R_{x = k}$ reflects a figure over the vertical - line $x = k$ and $R_{y=k}$ reflects a figure over the horizontal - line $y = k$. Rotation $r(\theta,O)$ rotates a figure $\theta$ degrees counter - clockwise about the origin $O$.

Step2: Observe the orientation and position change

The orientation of $\triangle PQR$ and $\triangle XYZ$ is the same, so there is no rotation. $\triangle PQR$ needs to be translated and then reflected. The $x$ - coordinates of the vertices of $\triangle PQR$ need to be increased by 6 to match the horizontal position of $\triangle XYZ$, and then reflected over the line $y=-2$.

Step3: Check the composition of rigid motions

The composition $R_{y = - 2}\circ T_{<6,0>}$ first translates $\triangle PQR$ 6 units to the right ($T_{<6,0>}$) and then reflects it over the line $y =-2$ ($R_{y=-2}$), which maps $\triangle PQR$ to $\triangle XYZ$.

Answer:

$R_{y = - 2}\circ T_{<6,0>}$