# Points A and B are at (4 ,1 ) and (7 ,5 ), respectively. Point A is rotated counterclockwise about the origin by pi/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?

May 23, 2018

$C = \left(\frac{15}{4} , \frac{7}{2}\right)$

#### Explanation:

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

• " a point "(x,y)to(-y,x)

$A \left(4 , 1\right) \to A ' \left(- 1 , 4\right) \text{ where A' is the image of A}$

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

$\text{expressing in terms of position vectors gives}$

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

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

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

$\textcolor{w h i t e}{\frac{1}{2} \underline{c}} = \left(\begin{matrix}7 \\ 5\end{matrix}\right) - \frac{1}{2} \left(\begin{matrix}- 1 \\ 4\end{matrix}\right)$

$\textcolor{w h i t e}{\frac{1}{2} \underline{c}} = \left(\begin{matrix}7 \\ 5\end{matrix}\right) - \left(\begin{matrix}- \frac{1}{2} \\ - 2\end{matrix}\right) = \left(\begin{matrix}\frac{15}{2} \\ 7\end{matrix}\right)$

$\underline{c} = \frac{1}{2} \left(\begin{matrix}\frac{15}{2} \\ 7\end{matrix}\right) = \left(\begin{matrix}\frac{15}{4} \\ \frac{7}{2}\end{matrix}\right)$

$\text{thus } C = \left(\frac{15}{4} , \frac{7}{2}\right)$