To determine the coordinates of the point [tex]\(H\)[/tex] after it undergoes the translation given by the rule [tex]\(T_{-5,9}(x, y)\)[/tex], we can follow these steps:
1. Identify the original coordinates of point [tex]\(H\)[/tex]:
The pre-image of point [tex]\(H\)[/tex] is given as [tex]\((-2, 3)\)[/tex].
2. Understand the translation rule [tex]\(T_{-5,9}\)[/tex]:
This translation rule means we need to move the point [tex]\(-5\)[/tex] units in the horizontal direction (left) and [tex]\(9\)[/tex] units in the vertical direction (up).
3. Apply the translation to the x-coordinate:
- The original [tex]\(x\)[/tex]-coordinate of [tex]\(H\)[/tex] is [tex]\(-2\)[/tex].
- According to the rule [tex]\(T_{-5,9}\)[/tex], we need to subtract [tex]\(5\)[/tex] from the [tex]\(x\)[/tex]-coordinate: [tex]\(-2 - 5 = -7\)[/tex].
4. Apply the translation to the y-coordinate:
- The original [tex]\(y\)[/tex]-coordinate of [tex]\(H\)[/tex] is [tex]\(3\)[/tex].
- According to the rule [tex]\(T_{-5,9}\)[/tex], we need to add [tex]\(9\)[/tex] to the [tex]\(y\)[/tex]-coordinate: [tex]\(3 + 9 = 12\)[/tex].
5. Combine the new coordinates:
After translation, the coordinates of point [tex]\(H\)[/tex] will be [tex]\((-7, 12)\)[/tex].
Hence, the new coordinates of point [tex]\(H\)[/tex] after the translation are [tex]\((-7, 12)\)[/tex], but none of the given options [tex]\((7, -8)\)[/tex], [tex]\((-7, 6)\)[/tex], [tex]\((3, -12)\)[/tex], [tex]\((2, 1)\)[/tex] match the result of the translation.
Therefore, there appears to be a mistake in the provided answer choices. The correct coordinates should be [tex]\((-7, 12)\)[/tex], but this is not listed among the provided options.
