To solve this problem, consider the original coordinates of point [tex]\( D \)[/tex] and the translation values that will be applied to it.
1. Identify the original coordinates of [tex]\( D \)[/tex]:
The original coordinates of [tex]\( D \)[/tex] are [tex]\( (-9, -6) \)[/tex].
2. Determine the translation transformation:
The translation transformation given is [tex]\( (x, y) \rightarrow (x+6, y-10) \)[/tex].
This means that you will add 6 units to the x-coordinate and subtract 10 units from the y-coordinate of point [tex]\( D \)[/tex].
3. Apply the translation to the [tex]\( x \)[/tex]-coordinate:
The original [tex]\( x \)[/tex]-coordinate of [tex]\( D \)[/tex] is [tex]\( -9 \)[/tex].
After applying the translation, the new [tex]\( x \)[/tex]-coordinate will be:
[tex]\[
x' = -9 + 6 = -3
\][/tex]
4. Apply the translation to the [tex]\( y \)[/tex]-coordinate:
The original [tex]\( y \)[/tex]-coordinate of [tex]\( D \)[/tex] is [tex]\( -6 \)[/tex].
After applying the translation, the new [tex]\( y \)[/tex]-coordinate will be:
[tex]\[
y' = -6 - 10 = -16
\][/tex]
Therefore, after applying the translation, the new coordinates of [tex]\( D \)[/tex] (which we call [tex]\( D' \)[/tex]) are [tex]\( (-3, -16) \)[/tex].
So the coordinates of [tex]\( D' \)[/tex] are:
[tex]\[ \boxed{(-3, -16)} \][/tex]
It's important to verify that this option is among the given choices. The choices provided are:
- [tex]\((-5, -2)\)[/tex]
- [tex]\((1, -12)\)[/tex]
- [tex]\((4, -15)\)[/tex]
- [tex]\((-9, -6)\)[/tex]
The coordinates [tex]\( (-3, -16) \)[/tex] are not listed in the given choices, which might suggest that there is an error in option listing or choices selection. However, based on the translation provided and initial coordinates, the correct calculated coordinates for [tex]\( D' \)[/tex] are indeed [tex]\( (-3, -16) \)[/tex].
