Points A and B are at $(9 ,3 )$ and $(4 ,8 )$ , respectively. Point A is rotated counterclockwise about the origin by $pi$ and dilated about point C by a factor of $3$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-07-06T12:06:17

The coordinates of point $C$ are $=(-31/2,-17/2)$

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

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

Therefore, the transformation of point $A$ is

$A'=((-1,0),(0,-1))((9),(3))=((-9),(-3))$

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

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

$((4-x),(8-y))=3((-9-x),(-3-y))$

So,

$4-x=3(-9-x)$

$4-x=-27-3x$

$2x=-31$

$x=-31/2$

and

$8-y=3(-3-y)$

$8-y=-9-3y$

$2y=-17$

$y=-17/2$

Therefore,

The point $C=(-31/2,-17/2)$

Points A and B are at (9 ,3 )(9 ,3 ) and (4 ,8 )(4 ,8 ), respectively. Point A is rotated counterclockwise about the origin by pi pi 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?