Points A and B are at $(4, 9)$ $(4 ,9 )$ and $(6, 8)$ $(6 ,8 )$ , respectively. Point A is rotated counterclockwise about the origin by $\frac{3 π}{2}$ $(3pi)/2$ 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 · 2017-06-15T18:37:26

The point $C = (10, - 8)$ $C=(10,-8)$

The matrix of a rotation counterclockwise by $\frac{3}{2} π$ $3/2pi$ about the origin is

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

Therefore, the transformation of point $A$ is

$A'=((0,1),(-1,0))((4),(9))=((9),(-4))$

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

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

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

So,

$6-x=3(9-x)$

$6-x=36-4x$

$3x=30$

$x=10$

and

$8-y=4(-4-y)$

$8-y=-16-4y$

$3y=-24$

$y=-8$

Therefore,

point $C=(10,-8)$

Points A and B are at (4 ,9 )(4,9)(4 ,9 ) and (6 ,8 )(6,8)(6 ,8 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2 3π2(3pi)/2 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?