The rule [tex]\( T_{5,-0.5^{\circ}}{ }^0 R_{0,180}(x, y) \)[/tex] is applied to [tex]\( \Delta FGH \)[/tex] to produce [tex]\( \triangle F'G'H' \)[/tex].

What are the coordinates of vertex [tex]\( F'' \)[/tex] of [tex]\( \Delta F''G'H'' \)[/tex]?

A. (4, -1.5)
B. (4, -0.5)
C. (-1.5, 4)
D. (-0.5, 4)



Answer :

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]