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 $π$ $pi$ 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?

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Answer 1 · 2017-07-06T12:06:17

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

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

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

Therefore, the transformation of point $A$ is

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

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