# Points A and B are at (4 ,6 ) and (7 ,5 ), respectively. Point A is rotated counterclockwise about the origin by pi/2  and dilated about point C by a factor of 5 . If point A is now at point B, what are the coordinates of point C?

Jan 19, 2018

$\left(- \frac{37}{4} , \frac{15}{4}\right)$

#### Explanation:

Rotating Point A counterclockwise by $\frac{\pi}{2}$ will give you a point at A' $\left(- 6 , 4\right)$

The distance between $A ' : \left(- 6 , 4\right) \mathmr{and} B : \left(7 , 5\right)$ is $\sqrt{{13}^{2} + {1}^{2}} = \sqrt{170}$

The distance between A' and B must be five times the distance between A' and C

The vector to go from B to A' is $\left\{- 13 , - 1\right\}$

1/4 of that vector is $\left\{- \frac{13}{4} , - \frac{1}{4}\right\}$

Apply that to A' to get $\left(- 6 - \frac{13}{4} , 4 - \frac{1}{4}\right) = \left(- \frac{37}{4} , \frac{15}{4}\right)$

Jan 19, 2018

$C = \left(- \frac{37}{4} , \frac{15}{4}\right)$

#### Explanation:

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(4 , 6\right) \to A ' \left(- 6 , 4\right) \text{ where A' is the image of A}$

$\Rightarrow \vec{C B} = \textcolor{red}{5} \vec{C A '}$

$\Rightarrow \underline{b} - \underline{c} = 5 \left(\underline{a} ' - \underline{c}\right)$

$\Rightarrow \underline{b} - \underline{c} = 5 \underline{a} ' - 5 \underline{c}$

$\Rightarrow 4 \underline{c} = 5 \underline{a} ' - \underline{b}$

$\textcolor{w h i t e}{\Rightarrow 4 \underline{c}} = 5 \left(\begin{matrix}- 6 \\ 4\end{matrix}\right) - \left(\begin{matrix}7 \\ 5\end{matrix}\right)$

$\textcolor{w h i t e}{\Rightarrow \underline{c}} = \left(\begin{matrix}- 30 \\ 20\end{matrix}\right) - \left(\begin{matrix}7 \\ 5\end{matrix}\right) = \left(\begin{matrix}- 37 \\ 15\end{matrix}\right)$

$\Rightarrow \underline{c} = \frac{1}{4} \left(\begin{matrix}- 37 \\ 15\end{matrix}\right) = \left(\begin{matrix}- \frac{37}{4} \\ \frac{15}{4}\end{matrix}\right)$

$\Rightarrow C = \left(- \frac{37}{4} , \frac{15}{4}\right)$