Points A and B are at $(9, 9)$ $(9 ,9 )$ and $(7, 8)$ $(7 ,8 )$ , respectively. Point A is rotated counterclockwise about the origin by $π$ $pi$ and dilated about point C by a factor of $2$ $2$ . If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2018-03-02T00:49:04

Point $C (- 25, - 26)$ $C(-25, -26)$

From $A (9, 9)$ $A(9, 9)$
point A will be at $A'(x_a', y_a')=(-9, -9)$ after rotation of $pi$ whether by counterclockwise or clockwise.

The formula for dilation with center of dilation $C(x_c, y_c)$ by a factor $k$ is
$x_a''=k(x_a'-x_c)+x_c$ and $y_a''=k(y_a'-y_c)+y_c$

where $(x_a'', y_a'')$ is the coordinates of the final position of $A$

Given that $B$ is at $(7, 8)$ , therefore $x_a''=7$ and $y_a''=8$

The coordinates of $C(x_c, y_c)$ can now be computed with $k=2$
$x_a''=k(x_a'-x_c)+x_c$
$7=2(-9-x_c)+x_c$
$7=-18-2*x_c+x_c$
$7=-18-x_c$
$x_c=-25$

And
$y_a''=k(y_a'-y_c)+y_c$
$8=2(-9-y_c)+y_c$
$8=-18-2y_c+y_c$
$8=-18-y_c$
$y_c=-26$

$C(x_c, y_c)=(-25, -26)$

God bless....I hope the explanation is useful.

Points A and B are at (9 ,9 )(9,9)(9 ,9 ) and (7 ,8 )(7,8)(7 ,8 ), respectively. Point A is rotated counterclockwise about the origin by pi πpi and dilated about point C by a factor of 2 22 . If point A is now at point B, what are the coordinates of point C?