Question

the line segment xy with endpoints x(3, 1) and y(2, -2) is rotated 90° counterclockwise about (-6, 4). what are the endpoints of xy?

Explanation:

Step1: Translate points relative to center

To rotate a point $(x,y)$ counter - clockwise 90° about a center $(a,b)$, first translate the point so that the center is at the origin. For point $X(3,1)$ and center $(-6,4)$:
$x_{trans}=3 - (-6)=9$
$y_{trans}=1 - 4=-3$
For point $Y(2,-2)$ and center $(-6,4)$:
$x_{trans}=2-(-6)=8$
$y_{trans}=-2 - 4=-6$

Step2: Apply 90° counter - clockwise rotation rule

The rule for a 90° counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$.
For the translated point of $X$:
$x_{rotated}=-(-3)=3$
$y_{rotated}=9$
For the translated point of $Y$:
$x_{rotated}=-(-6)=6$
$y_{rotated}=8$

Step3: Translate back to original position

Add the coordinates of the center $(-6,4)$ back to the rotated - translated points.
For $X'$:
$x = 3+(-6)=-3$
$y = 9 + 4=13$
For $Y'$:
$x = 6+(-6)=0$
$y = 8 + 4=12$

Answer:

$X'(-3,13)$ and $Y'(0,12)$