Points A and B are at $(5, 9)$ $(5 ,9 )$ and $(8, 6)$ $(8 ,6 )$ , respectively. Point A is rotated counterclockwise about the origin by $π$ $pi$ and dilated about point C by a factor of $4$ $4$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-07-05T08:29:40

The point $C = (- \frac{28}{3}, - 14)$ $C=(-28/3, -14)$

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

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

Therefore, the transformation of point $A$ is

$A'=((-1,0),(0,-1))((5),(9))=((-5),(-9))$

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

$vec(CB)=4vec(CA')$

$((8-x),(6-y))=4((-5-x),(-9-y))$

So,

$8-x=4(-5-x)$

$8-x=-20-4x$

$3x=-28$

$x=-28/3$

and

$6-y=4(-9-y)$

$6-y=-36-4y$

$3y=-42$

$y=-14$

Therefore,

The point $C=(-28/3,-14)$

Points A and B are at (5 ,9 )(5,9)(5 ,9 ) and (8 ,6 )(8,6)(8 ,6 ), respectively. Point A is rotated counterclockwise about the origin by pi πpi and dilated about point C by a factor of 4 44 . If point A is now at point B, what are the coordinates of point C?