Points A and B are at $(8, 3)$ $(8 ,3 )$ and $(5, 7)$ $(5 ,7 )$ , respectively. Point A is rotated counterclockwise about the origin by $\frac{π}{2}$ $pi/2$ 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 · 2017-06-23T11:41:42

The point $C = (- 7, \frac{17}{2})$ $C=(-7,17/2)$

The matrix of a rotation counterclockwise by $\frac{1}{2} π$ $1/2pi$ about the origin is

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

Therefore, the transformation of point $A$ is

$A'=((0,-1),(1,0))((8),(3))=((-3),(8))$

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

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

$((5-x),(7-y))=3((-3-x),(8-y))$

So,

$5-x=3(-3-x)$

$5-x=-9-3x$

$2x=-14$

$x=-7$

and

$7-y=3(8-y)$

$7-y=24-3y$

$2y=24-7$

$y=17/2$

Therefore,

point $C=(-7,17/2)$

Points A and B are at (8 ,3 )(8,3)(8 ,3 ) and (5 ,7 )(5,7)(5 ,7 ), respectively. Point A is rotated counterclockwise about the origin by pi/2 π2pi/2 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?