Points A and B are at $(3 ,5 )$ and $(6 ,1 )$ , respectively. Point A is rotated counterclockwise about the origin by $(3pi)/2$ and dilated about point C by a factor of $3$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-06-08T08:59:48

The coordinates of point $C=(9/2,-5)$

The matrix of a rotation counterclockwise by $3/2pi$ about the origin is

$((0,1),(-1,0))$

Therefore, the trasformation of point $A$ is

$A'=((0,1),(-1,0))((3),(5))=((5),(-3))$

Let point $C$ be $(x,y)$ , then

$vec(CB)=3 vec(CA')$

$((6-x),(1-y))=3((5-x),(-3-y))$

So,

$6-x=3(5-x)$

$15-3x=6-x$

$2x=9$

$x=9/2$

and

$1-y=3(-3-y)$

$-9-3y=1-y$

$2y=-10$

$y=-5$

Therefore,

point $C=(9/2,-5)$

Points A and B are at (3 ,5 )(3 ,5 ) and (6 ,1 )(6 ,1 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2 (3pi)/2 and dilated about point C by a factor of 3 3 . If point A is now at point B, what are the coordinates of point C?