# Points A and B are at (2 ,5 ) and (6 ,2 ), respectively. Point A is rotated counterclockwise about the origin by (3pi)/2  and dilated about point C by a factor of 1/2 . If point A is now at point B, what are the coordinates of point C?

Jun 8, 2018

color(red)("Coordinates of " C (7,6)

#### Explanation:

$A \left(2 , 5\right) , B \left(6 , 2\right) , \text{rotation }$(3pi)/2$, \text{dilation factor} \frac{1}{2}$

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

$A \left(2 , 5\right) \rightarrow A ' \left(5 , - 2\right)$

$\vec{B C} = \left(\frac{1}{2}\right) \vec{A ' C}$

$b - c = \left(\frac{1}{2}\right) a ' - \left(\frac{1}{2}\right) c$

$\left(\frac{1}{2}\right) c = - \left(\frac{1}{2}\right) a ' + b$

$\left(\frac{1}{2}\right) C \left(\begin{matrix}x \\ y\end{matrix}\right) = - \left(\frac{1}{2}\right) \left(\begin{matrix}5 \\ - 2\end{matrix}\right) + \left(\begin{matrix}6 \\ 2\end{matrix}\right) = \left(\begin{matrix}\frac{7}{2} \\ 3\end{matrix}\right)$

color(red)("Coordinates of " 2 *C ((7/2),3) = C(7,6)

Jun 8, 2018

$C = \left(7 , 6\right)$

#### Explanation:

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

• " a point "(x,y)to(y,-x)

$A \left(2 , 5\right) \to A ' \left(5 , - 2\right) \text{ where A' is the image of A}$

$\vec{C B} = \textcolor{red}{\frac{1}{2}} \vec{C A '}$

$\underline{b} - \underline{c} = \frac{1}{2} \left(\underline{a} ' - \underline{c}\right)$

$\underline{b} - \underline{c} = \frac{1}{2} \underline{a} ' - \frac{1}{2} \underline{c}$

$\frac{1}{2} \underline{c} = \underline{b} - \frac{1}{2} \underline{a} '$

$\textcolor{w h i t e}{\frac{1}{2} \underline{c}} = \left(\begin{matrix}6 \\ 2\end{matrix}\right) - \frac{1}{2} \left(\begin{matrix}5 \\ - 2\end{matrix}\right)$

$\textcolor{w h i t e}{\frac{1}{2} \underline{c}} = \left(\begin{matrix}6 \\ 2\end{matrix}\right) - \left(\begin{matrix}\frac{5}{2} \\ - 1\end{matrix}\right) = \left(\begin{matrix}\frac{7}{2} \\ 3\end{matrix}\right)$

$\underline{c} = 2 \left(\begin{matrix}\frac{7}{2} \\ 3\end{matrix}\right) = \left(\begin{matrix}7 \\ 6\end{matrix}\right)$

$\Rightarrow C = \left(7 , 6\right)$