To solve this problem, we need to understand the transformation given by the rule [tex]$T_{5,-0.5^\circ}{}^0 R_{0,180}$[/tex]. This rule combines a rotation and a translation, and we will focus on understanding each part step by step.
1. Understanding [tex]\( R_{0,180} \)[/tex]:
- Considering the notation [tex]$R_{0,180}$[/tex], this represents a rotation of [tex]$180^\circ$[/tex] about the origin [tex]\((0,0)\)[/tex].
- Rotating point [tex]\((x, y)\)[/tex] by [tex]$180^\circ$[/tex] about the origin results in the point [tex]\((-x, -y)\)[/tex].
2. Initial Coordinates:
- The initial coordinates of vertex [tex]\( F \)[/tex] are [tex]\((4, -1.5)\)[/tex].
3. Applying the Rotation [tex]\(R_{0,180}\)[/tex]:
- Applying the [tex]$180^\circ$[/tex] rotation to [tex]\((4, -1.5)\)[/tex] involves negating both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates.
- Therefore, the coordinates of vertex [tex]\(F''''\)[/tex] after the rotation will be:
[tex]\[
(-4, 1.5)
\][/tex]
Thus, the coordinates of vertex [tex]\( F^{\prime\prime} \)[/tex] of [tex]\(\triangle F^{\prime\prime} G^{\prime} H^{\prime\prime}\)[/tex] are:
[tex]\[
(-4, 1.5)
\][/tex]
