Let's solve this step-by-step using the given transformations. We're going to apply the transformations to the point [tex]\(F\)[/tex] with initial coordinates [tex]\((4, -1.5)\)[/tex].
First, we need to rotate the point [tex]\( (4, -1.5) \)[/tex] 180 degrees about the origin. A 180-degree rotation about the origin transforms a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
- Original coordinates: [tex]\((4, -1.5)\)[/tex]
- Rotate 180 degrees: [tex]\((-4, 1.5)\)[/tex]
Next, we translate the resulting point [tex]\((-4, 1.5)\)[/tex] 5 units to the right and [tex]\(-0.5\)[/tex] units down.
- Translating 5 units to the right: [tex]\((-4 + 5, 1.5)\)[/tex] = [tex]\((1, 1.5)\)[/tex]
- Translating [tex]\(-0.5\)[/tex] units down: [tex]\((1, 1.5 - 0.5)\)[/tex] = [tex]\((1, 1.0)\)[/tex]
Therefore, the coordinates of vertex [tex]\( F^{\prime \prime} \)[/tex] of [tex]\( \Delta F^{\prime \prime} G^{\prime \prime} H^{\prime\prime} \)[/tex] are [tex]\((1, 1.0)\)[/tex].
