To find the new coordinates of point [tex]\( C \)[/tex] after rotating the triangle [tex]\( ABC \)[/tex] [tex]\( 90^{\circ} \)[/tex] clockwise about the point [tex]\( (4, 1) \)[/tex], we can follow these steps:
1. Translate Point C to the Origin (Relative to the Rotation Point):
We first translate the coordinates of point [tex]\( C(4, 2) \)[/tex] so that the point [tex]\( (4,1) \)[/tex] becomes the origin.
The translated coordinates [tex]\( C' \)[/tex] would be:
[tex]\[
C' = (C_x - P_x, C_y - P_y)
\][/tex]
Plugging in the values:
[tex]\[
C' = (4 - 4, 2 - 1) = (0, 1)
\][/tex]
2. Apply the 90-Degree Clockwise Rotation:
For a [tex]\( 90^{\circ} \)[/tex] clockwise rotation about the origin, the transformation is given by:
[tex]\[
(x', y') \to (y', -x')
\][/tex]
Applying this to [tex]\( C'(0, 1) \)[/tex]:
[tex]\[
C'' = (1, 0)
\][/tex]
3. Translate Back to the Original Center (4,1):
After the rotation, we must translate the result back to the original coordinate system centered at [tex]\( (4, 1) \)[/tex].
The final coordinates of [tex]\( C \)[/tex] would be:
[tex]\[
C'' = (C''_x + P_x, C''_y + P_y)
\][/tex]
Plugging in the values:
[tex]\[
C'' = (1 + 4, 0 + 1) = (5, 1)
\][/tex]
Therefore, the new coordinates of [tex]\( C \)[/tex] after the rotation are [tex]\( (5, 1) \)[/tex].
The correct answer is:
c. [tex]\( (5, 1) \)[/tex]
