Points A and B are at $(2, 4)$ $(2 ,4 )$ and $(7, 2)$ $(7 ,2 )$ , 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 $2$ $2$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-06-24T06:26:14

The coordinates of point $C$ $C$ are $(1, - 6)$ $(1,-6)$

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

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

Therefore, the transformation of point $A$ is

$A'=((0,1),(-1,0))((2),(4))=((4),(-2))$

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

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

$((7-x),(2-y))=2((4-x),(-2-y))$

So,

$7-x=2(4-x)$

$7-x=8-2x$

$x=1$

and

$2-y=2(-2-y)$

$2-y=-4-2y$

$y=-4-2$

$y=-6$

Therefore,

point $C=(1,-6)$

Points A and B are at (2 ,4 )(2,4)(2 ,4 ) and (7 ,2 )(7,2)(7 ,2 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2 3π2(3pi)/2 and dilated about point C by a factor of 2 22 . If point A is now at point B, what are the coordinates of point C?