Points A and B are at $(4 ,9 )$ and $(7 ,2 )$ , respectively. Point A is rotated counterclockwise about the origin by $(3pi)/2$ and dilated about point C by a factor of $1/2$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-05-07T08:32:24

The point $C$ is $(-5/3,8)$

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

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

Therefore, the trasformation of point $A$ is

$A'=((0,1),(-1,0))((4),(9))=((9),(-4))$

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

$vec(CB)=1/2 vec(CA')$

$((7-x),(2-y))=1/2((9-x),(-4-y))$

So,

$7-x=1/2(9-x)$

$14-2x=9-5x$

$3x=9-14=-5$

$x=-5/3$

and

$2-y=1/2(-4-y)$

$4-2y=-4-y$

$y=8$

Therefore,

point $C=(-5/3,8)$

Points A and B are at (4 ,9 )(4 ,9 ) and (7 ,2 )(7 ,2 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2 (3pi)/2 and dilated about point C by a factor of 1/2 1/2 . If point A is now at point B, what are the coordinates of point C?