# Point A is at (9 ,5 ) and point B is at (2 ,4 ). Point A is rotated pi  clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

Jul 6, 2017

The new coordinates are $\left(- 9 , - 5\right)$ and the distance has changed by $= 7.14$

#### Explanation:

The matrix of a rotation clockwise by $\pi$ about the origin is

$\left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right)$

Therefore, the transformation of point $A$ is

$A ' = \left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right) \left(\begin{matrix}9 \\ 5\end{matrix}\right) = \left(\begin{matrix}- 9 \\ - 5\end{matrix}\right)$

The distance $A B$ is

$A B = \sqrt{{\left(2 - 9\right)}^{2} + {\left(4 - 5\right)}^{2}}$

$= \sqrt{49 + 1}$

$= \sqrt{50}$

The distance $A ' B$ is

$A ' B = \sqrt{{\left(2 - \left(- 9\right)\right)}^{2} + {\left(4 - \left(- 5\right)\right)}^{2}}$

$= \sqrt{121 + 81}$

$= \sqrt{202}$

The distance has changed by

$= \sqrt{202} - \sqrt{50}$

$= 7.14$