Points A and B are at $(3 ,9 )$ and $(9 ,2 )$ , respectively. Point A is rotated counterclockwise about the origin by $(3pi)/2$ and dilated about point C by a factor of $2$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-03-28T07:12:49

The coordinates of $C=(-3,-20)$

The point $A=(3,9)$ becomes (after a rotation of $3/2pi$ )

$A'=(3,-9)$

Let the center of the dilation be $C=(x,y)$

$B=(9,2)$

We perform this with vectors

$vecCB=2vec(CA')$

$<9-x,2-y> = 2<3-x,-9-y>$

Therefore,

$9-x=2(3-x)=6-2x$

$2x-x=6-9$

$x=-3$

and

$2-y=2(-9-y)=-18-2y$

$2y-y=-18-2$

$y=-20$

So,

$C=(-3,-20)$

Points A and B are at (3 ,9 )(3 ,9 ) and (9 ,2 )(9 ,2 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2 (3pi)/2 and dilated about point C by a factor of 2 2 . If point A is now at point B, what are the coordinates of point C?