# Points A and B are at (4 ,7 ) and (3 ,9 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  and dilated about point C by a factor of 4 . If point A is now at point B, what are the coordinates of point C?

Jul 22, 2018

color(indigo)("Coordinates of "C((x),(y)) = ((19),(-43))

#### Explanation:

$A \left(4 , 7\right) , B \left(3 , 9\right) , \text{counterclockwise rotation }$(3pi)/2$, \text{dilation factor} 4$

New coordinates of A after $\frac{3 \pi}{2}$ counterclockwise rotation

$A \left(4 , 7\right) \rightarrow A ' \left(7 , - 4\right)$

$\vec{B C} = \left(4\right) \vec{A ' C}$

$b - c = \left(4\right) a ' - \left(4\right) c$

$c = \left(4\right) a ' - \left(3\right) b$

color(indigo)(C((x),(y)) = (4)((7),(-4)) - (3)((3),(9)) = ((19),(-43))

Jul 23, 2018

$C = \left(\frac{25}{3} , - \frac{25}{3}\right)$

#### Explanation:

$\text{under a counterclockwise rotation about the origin of } \frac{3 \pi}{2}$

• " a point "(x,y)to(y,-x)

$A \left(4 , 7\right) \to A ' \left(7 , - 4\right) \text{ where A' is the image of A}$

$\vec{C B} = \textcolor{red}{4} \vec{C A '}$

$\underline{b} - \underline{c} = 4 \left(\underline{a} ' - \underline{c}\right)$

$\underline{b} - \underline{c} = 4 \underline{a} ' - 4 \underline{c}$

$3 \underline{c} = 4 \underline{a} ' - \underline{b}$

$\textcolor{w h i t e}{3 \underline{c}} = 4 \left(\begin{matrix}7 \\ - 4\end{matrix}\right) - \left(\begin{matrix}3 \\ 9\end{matrix}\right)$

$\textcolor{w h i t e}{3 \underline{c}} = \left(\begin{matrix}28 \\ - 16\end{matrix}\right) - \left(\begin{matrix}3 \\ 9\end{matrix}\right) = \left(\begin{matrix}25 \\ - 25\end{matrix}\right)$

$\underline{c} = \frac{1}{3} \left(\begin{matrix}25 \\ - 25\end{matrix}\right) = \left(\begin{matrix}\frac{25}{3} \\ - \frac{25}{3}\end{matrix}\right)$

$\Rightarrow C = \left(\frac{25}{3} , - \frac{25}{3}\right)$