Question

the rule $t_{5, - 0.5}circ r_{0,180^{circ}}(x,y)$ is applied to $\triangle fgh$ to produce $\triangle fgh$. what are the coordinates of vertex $f$ of $\triangle fgh$? (4, -1.5) (4, -0.5) (-1.5, 4) (-0.5, 4)

Explanation:

Step1: Analyze the rotation rule

The rotation $R_{0,180^{\circ}}(x,y)=(-x,-y)$. Assume the original coordinates of $F$ are $(x_0,y_0)$. After rotation, the coordinates become $(-x_0,-y_0)$.

Step2: Analyze the translation rule

The translation $T_{5, - 0.5}(x,y)=(x + 5,y-0.5)$. After rotation, if the new - coordinates are $(x_1,y_1)=(-x_0,-y_0)$, then after translation, the final coordinates $(x_2,y_2)=(x_1 + 5,y_1-0.5)=(-x_0 + 5,-y_0-0.5)$.
From the graph, assume the original coordinates of $F$ are $(1,1)$.
After rotation $R_{0,180^{\circ}}$, the coordinates of $F$ become $(-1,-1)$.
After translation $T_{5,-0.5}$, we have $x=-1 + 5=4$ and $y=-1-0.5=-1.5$.

Answer:

$(4,-1.5)$