Points A and B are at $(9, 9)$ $(9 ,9 )$ and $(7, 6)$ $(7 ,6 )$ , respectively. Point A is rotated counterclockwise about the origin by $π$ $pi$ 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 · 2017-06-23T17:27:08

The point $C = (- 25, - 24)$ $C=(-25,-24)$

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

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

Therefore, the transformation of point $A$ is

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

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

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

$((7-x),(6-y))=2((-9-x),(-9-y))$

So,

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

$7-x=-18-2x$

$x=-25$

and

$6-y=2(-9-y)$

$6-y=-18-2y$

$y=-18-6$

$y=-24$

Therefore,

The point $C=(-25,-24)$

Points A and B are at (9 ,9 )(9,9)(9 ,9 ) and (7 ,6 )(7,6)(7 ,6 ), respectively. Point A is rotated counterclockwise about the origin by pi πpi 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?