Points A and B are at $(3, 5)$ $(3 ,5 )$ 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 · 2017-06-06T05:21:53

The coordinates of point are $C = (\frac{13}{2}, - 7)$ $C=(13/2,-7)$

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

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

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

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

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

So,

$2-x=3(5-x)$

$15-3x=2-x$

$2x=13$

$x=13/2$

and

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

$-9-3y=5-y$

$2y=-14$

$y=-7$

Therefore,

point $C=(13/2,-7)$

Points A and B are at (3 ,5 )(3,5)(3 ,5 ) 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?