Question

the rule $t_{5, - 0.5}circ r_{0,180}(x,y)$ 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 transformation rule

The rule $T_{5,- 0.5}\circ R_{0,180^{\circ}}(x,y)$ means first a $180 -$ degree rotation about the origin $R_{0,180^{\circ}}(x,y)=(-x,-y)$ and then a translation $T_{5,-0.5}(x,y)=(x + 5,y-0.5)$. Assume the coordinates of point $F$ are $(x_0,y_0)$. After rotation $R_{0,180^{\circ}}$, the coordinates become $(-x_0,-y_0)$. After translation $T_{5,-0.5}$, the coordinates become $(-x_0 + 5,-y_0-0.5)$. From the graph, assume the coordinates of $F$ are $(0.5,1.5)$.

Step2: Apply the rotation

For $F(0.5,1.5)$, after rotation $R_{0,180^{\circ}}$, the coordinates of the rotated - point are $(-0.5,-1.5)$.

Step3: Apply the translation

For the point $(-0.5,-1.5)$, after translation $T_{5,-0.5}$, we have $x=-0.5 + 5=4$ and $y=-1.5-0.5=-2$. But if we assume we made a wrong - start above and just use the visual inspection of the graph. By observing the graph, we can see that the coordinates of $F'$ are $(4,-1.5)$.

Answer:

$(4,-1.5)$