# Point A is at (-2 ,5 ) and point B is at (-3 ,8 ). 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?

Jun 7, 2017

The new point is $= \left(2 , - 5\right)$ and the distance has changed by $= 12.93$

#### Explanation:

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

$= \left(\begin{matrix}\cos \left(- \pi\right) & - \sin \left(- \pi\right) \\ \sin \left(- \pi\right) & \cos \left(- \pi\right)\end{matrix}\right) = \left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right)$

Therefore, the trasformation of point $A$ into $A '$ is

$A ' = \left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right) \left(\begin{matrix}- 2 \\ 5\end{matrix}\right) = \left(\begin{matrix}2 \\ - 5\end{matrix}\right)$

Distance $A B$ is

$= \sqrt{{\left(- 3 - \left(- 2\right)\right)}^{2} + {\left(8 - \left(5\right)\right)}^{2}}$

$= \sqrt{1 + 9}$

$= \sqrt{10}$

Distance $A ' B$ is

$= \sqrt{{\left(- 3 - \left(2\right)\right)}^{2} + {\left(8 - \left(- 5\right)\right)}^{2}}$

$= \sqrt{25 + 169}$

$= \sqrt{194}$

The distance has changed by

$= \sqrt{194} - \sqrt{10}$

$= 12.93$