Answer :
Sure, let's determine the new location of point [tex]\( V \)[/tex] after the given transformations.
### Step-by-Step Solution:
1. Starting Coordinates:
The original coordinates of point [tex]\( V \)[/tex] are [tex]\((3, -9)\)[/tex].
2. Translation:
The translation rule provided is [tex]\((x, y) \rightarrow (x-2, y-4)\)[/tex].
- Apply this translation to the original coordinates of [tex]\( V \)[/tex]:
- New [tex]\( x \)[/tex]-coordinate: [tex]\( 3 - 2 = 1 \)[/tex]
- New [tex]\( y \)[/tex]-coordinate: [tex]\( -9 - 4 = -13 \)[/tex]
- So, after the translation, the new coordinates of [tex]\( V \)[/tex] are [tex]\((1, -13)\)[/tex].
3. Rotation 90 Degrees Counterclockwise:
The rule for rotating a point [tex]\((x, y)\)[/tex] 90 degrees counterclockwise is that the new coordinates are [tex]\((-y, x)\)[/tex].
- Apply this rotation to the translated coordinates [tex]\((1, -13)\)[/tex]:
- New [tex]\( x \)[/tex]-coordinate: [tex]\(-(-13) = 13 \)[/tex]
- New [tex]\( y \)[/tex]-coordinate: [tex]\( 1 \)[/tex]
- Therefore, the coordinates of [tex]\( V \)[/tex] after this rotation are [tex]\((13, 1)\)[/tex].
4. Conclusion:
The new location of point [tex]\( V \)[/tex] after the translation followed by a 90-degree counterclockwise rotation is [tex]\((13, 1)\)[/tex].
Note: The provided multiple-choice options do not include the translated and rotated coordinates, suggesting a misunderstanding or misprint in the question. The correct transformed coordinates are indeed [tex]\((13, 1)\)[/tex].
### Step-by-Step Solution:
1. Starting Coordinates:
The original coordinates of point [tex]\( V \)[/tex] are [tex]\((3, -9)\)[/tex].
2. Translation:
The translation rule provided is [tex]\((x, y) \rightarrow (x-2, y-4)\)[/tex].
- Apply this translation to the original coordinates of [tex]\( V \)[/tex]:
- New [tex]\( x \)[/tex]-coordinate: [tex]\( 3 - 2 = 1 \)[/tex]
- New [tex]\( y \)[/tex]-coordinate: [tex]\( -9 - 4 = -13 \)[/tex]
- So, after the translation, the new coordinates of [tex]\( V \)[/tex] are [tex]\((1, -13)\)[/tex].
3. Rotation 90 Degrees Counterclockwise:
The rule for rotating a point [tex]\((x, y)\)[/tex] 90 degrees counterclockwise is that the new coordinates are [tex]\((-y, x)\)[/tex].
- Apply this rotation to the translated coordinates [tex]\((1, -13)\)[/tex]:
- New [tex]\( x \)[/tex]-coordinate: [tex]\(-(-13) = 13 \)[/tex]
- New [tex]\( y \)[/tex]-coordinate: [tex]\( 1 \)[/tex]
- Therefore, the coordinates of [tex]\( V \)[/tex] after this rotation are [tex]\((13, 1)\)[/tex].
4. Conclusion:
The new location of point [tex]\( V \)[/tex] after the translation followed by a 90-degree counterclockwise rotation is [tex]\((13, 1)\)[/tex].
Note: The provided multiple-choice options do not include the translated and rotated coordinates, suggesting a misunderstanding or misprint in the question. The correct transformed coordinates are indeed [tex]\((13, 1)\)[/tex].
