Points A and B are at $(9, 2)$ $(9 ,2 )$ and $(2, 5)$ $(2 ,5 )$ , respectively. Point A is rotated counterclockwise about the origin by $\frac{3 π}{2}$ $(3pi)/2$ 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?

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Answer 1 · 2017-06-02T09:46:39

The coordinates of the point $C = (2, - 16)$ $C=(2,-16)$

The matrix of a rotation counterclockwise by $\frac{3}{2} π$ $3/2pi$ about the origin is

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

Therefore, the trasformation of point $A$ is

$A'=((0,1),(-1,0))((9),(2))=((2),(-9))$

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

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

$((2-x),(5-y))=3((2-x),(-9-y))$

So,

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

$6-3x=2-x$

$2x=4$

$x=2$

and

$5-y=3(-9-y)$

$-27-3y=5-y$

$2y=-32$

$y=-16$

Therefore,

point $C=(2,-16)$

Points A and B are at (9 ,2 )(9,2)(9 ,2 ) and (2 ,5 )(2,5)(2 ,5 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2 3π2(3pi)/2 and dilated about point C by a factor of 3 33 . If point A is now at point B, what are the coordinates of point C?