Points A and B are at $(4 ,6 )$ and $(8 ,3 )$ , respectively. Point A is rotated counterclockwise about the origin by $pi$ 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 · 2017-07-12T08:50:32

The coordinates of point $C$ are $(20,12)$

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

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

Therefore, the transformation of point $A$ is

$A'=((-1,0),(0,-1))((4),(6))=((-4),(-6))$

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

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

$((8-x),(3-y))=1/2((-4-x),(-6-y))$

So,

$8-x=1/2(-4-x)$

$16-2x=-4-x$

$x=20$

and

$3-y=1/2(-6-y)$

$6-2y=-6-y$

$y=12$

Therefore,

point $C=(20,12)$

Points A and B are at (4 ,6 )(4 ,6 ) and (8 ,3 )(8 ,3 ), respectively. Point A is rotated counterclockwise about the origin by pi pi 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?