Points A and B are at $(5, 8)$ $(5 ,8 )$ and $(7, 1)$ $(7 ,1 )$ , respectively. Point A is rotated counterclockwise about the origin by $π$ $pi$ and dilated about point C by a factor of $5$ $5$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2017-05-03T17:04:50

The point $C$ $C$ is $= (- 8, - \frac{41}{4})$ $=(-8,-41/4)$

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

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

Therefore, the trasformation of point $A$ is

$A'=((-1,0),(0,-1))((5),(8))=((-5),(-8))$

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

$vec(CB)=5 vec(CA')$

$((7-x),(1-y))=5((-5-x),(-8-y))$

So,

$7-x=5(-5-x)$

$7-x=-25-5x$

$4x=-25-7=-32$

$x=-8$

and

$1-y=5(-8-y)$

$1-y=-40-5y$

$4y=-41$

$y=-41/4$

Therefore,

point $C=(-8,-41/4)$

Points A and B are at (5 ,8 )(5,8)(5 ,8 ) and (7 ,1 )(7,1)(7 ,1 ), respectively. Point A is rotated counterclockwise about the origin by pi πpi and dilated about point C by a factor of 5 55 . If point A is now at point B, what are the coordinates of point C?