# Points A and B are at (3 ,7 ) and (4 ,2 ), respectively. Point A is rotated counterclockwise about the origin by pi  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 13, 2018

After Point A is rotated counterclockwise about the origin by $\pi$, its new coordinates are $\left(- 3 , - 7\right)$.

The difference between the $x$ coordinates of Point A and B now is $4 - \left(- 3\right) = 7$ and $y$ coordinates: $2 - \left(- 7\right) = 9$

Since Point A was dilated about Point C by a factor of 5, we can find out by how much the coordinates change with each integer increase in factor.

For the $x$ coordinate:
$\frac{7}{4} = 1.75$
and $y$ coordinate:
$\frac{9}{4} = 2.25$

So for every integer increase in factor, the point moves 1.75 to the right and 2.25 upwards.

Point C is therefore
$\left(- 3 - 1.75 , - 7 - 2.25\right)$
$= \left(- 4.75 , - 9.25\right)$

Jan 13, 2018

$C = \left(- \frac{19}{4} , - \frac{37}{4}\right)$

#### Explanation:

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

• " a point "(x,y)to(-x,-y)

$\Rightarrow A \left(3 , 7\right) \to A ' \left(- 3 , - 7\right) \text{ where A' is the image of A}$

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

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

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

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

$\textcolor{w h i t e}{4 \underline{c} \times} = 5 \left(\begin{matrix}- 3 \\ - 7\end{matrix}\right) - \left(\begin{matrix}4 \\ 2\end{matrix}\right)$

$\textcolor{w h i t e}{\times \times} = \left(\begin{matrix}- 15 \\ - 35\end{matrix}\right) - \left(\begin{matrix}4 \\ 2\end{matrix}\right) = \left(\begin{matrix}- 19 \\ - 37\end{matrix}\right)$

$\Rightarrow \underline{c} = \frac{1}{4} \left(\begin{matrix}- 19 \\ - 37\end{matrix}\right) = \left(\begin{matrix}- \frac{19}{4} \\ - \frac{37}{4}\end{matrix}\right)$

$\Rightarrow C = \left(- \frac{19}{4} , - \frac{37}{4}\right)$