Given triangle [tex]$ABC$[/tex] with vertices at points [tex]$A(-1,3)$[/tex], [tex]$B(1,-1)$[/tex], and [tex]$C(4,2)$[/tex], if the triangle is rotated [tex]$90^{\circ}$[/tex] clockwise about the point [tex]$(4,1)$[/tex], find the new coordinates of [tex]$C$[/tex].

Select one:
a. [tex]$(5, -1)$[/tex]
b. [tex]$(-1, -9)$[/tex]
c. [tex]$(5, 1)$[/tex]
d. [tex]$(-5, 1)$[/tex]



Answer :

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]