Let's follow the steps given to find the location of point [tex]\( T \)[/tex] following the transformations indicated.
1. Translation: Point [tex]\( T \)[/tex] starts at coordinates [tex]\( (3, -4) \)[/tex].
The translation rule given is [tex]\((x, y) \rightarrow(x-2, y-4)\)[/tex].
- Calculate the new [tex]\( x \)[/tex]-coordinate:
[tex]\[
x_{\text{translated}} = 3 - 2 = 1
\][/tex]
- Calculate the new [tex]\( y \)[/tex]-coordinate:
[tex]\[
y_{\text{translated}} = -4 - 4 = -8
\][/tex]
So, after the translation, point [tex]\( T \)[/tex] is at [tex]\((1, -8)\)[/tex].
2. Rotation: Next, we rotate the translated point [tex]\(90^{\circ}\)[/tex] counterclockwise.
The rule for rotating a point [tex]\(90^{\circ}\)[/tex] counterclockwise is [tex]\((x, y) \rightarrow(-y, x)\)[/tex].
- Apply this rule to the translated coordinates:
[tex]\[
x_{\text{rotated}} = -(-8) = 8
\][/tex]
[tex]\[
y_{\text{rotated}} = 1
\][/tex]
Therefore, after the rotation, the new coordinates of point [tex]\( T \)[/tex] are [tex]\((8, 1)\)[/tex].
Hence, the location of [tex]\( T \)[/tex] after applying the translation and rotation is [tex]\((8, 1)\)[/tex].
Given this information, none of the provided multiple-choice answers [tex]\((3, -9)\)[/tex], [tex]\((3, -4)\)[/tex], [tex]\((-2, -4)\)[/tex], and [tex]\((-2, -9)\)[/tex] are correct. The correct coordinates are [tex]\((8, 1)\)[/tex].
