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If rectangle STUV is translated using the rule [tex]\((x, y) \rightarrow (x-2, y-4)\)[/tex] and then rotated [tex]\(90^{\circ}\)[/tex] counterclockwise, what is the location of [tex]\(V\)[/tex]?

A. [tex]\((3, -9)\)[/tex]
B. [tex]\((3, -4)\)[/tex]
C. [tex]\((-2, -4)\)[/tex]
D. [tex]\((-2, -9)\)[/tex]

Sagot :

GinnyAnsweridn
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].