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

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Mar 2, 2018

Point $C \left(- 25 , - 26\right)$

#### Explanation:

From $A \left(9 , 9\right)$
point A will be at $A ' \left({x}_{a} ' , {y}_{a} '\right) = \left(- 9 , - 9\right)$ after rotation of $\pi$ whether by counterclockwise or clockwise.

The formula for dilation with center of dilation $C \left({x}_{c} , {y}_{c}\right)$ by a factor $k$ is
${x}_{a} ' ' = k \left({x}_{a} ' - {x}_{c}\right) + {x}_{c}$ and ${y}_{a} ' ' = k \left({y}_{a} ' - {y}_{c}\right) + {y}_{c}$

where $\left({x}_{a} ' ' , {y}_{a} ' '\right)$ is the coordinates of the final position of $A$

Given that $B$ is at $\left(7 , 8\right)$, therefore ${x}_{a} ' ' = 7$ and ${y}_{a} ' ' = 8$

The coordinates of $C \left({x}_{c} , {y}_{c}\right)$ can now be computed with $k = 2$
${x}_{a} ' ' = k \left({x}_{a} ' - {x}_{c}\right) + {x}_{c}$
$7 = 2 \left(- 9 - {x}_{c}\right) + {x}_{c}$
$7 = - 18 - 2 \cdot {x}_{c} + {x}_{c}$
$7 = - 18 - {x}_{c}$
${x}_{c} = - 25$

And
${y}_{a} ' ' = k \left({y}_{a} ' - {y}_{c}\right) + {y}_{c}$
$8 = 2 \left(- 9 - {y}_{c}\right) + {y}_{c}$
$8 = - 18 - 2 {y}_{c} + {y}_{c}$
$8 = - 18 - {y}_{c}$
${y}_{c} = - 26$

$C \left({x}_{c} , {y}_{c}\right) = \left(- 25 , - 26\right)$

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

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