# Point A is at (5 ,2 ) and point B is at (2 ,-4 ). Point A is rotated (3pi)/2  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?

Feb 6, 2018

New coordinate of color(red)(A (-2,5)

Reduction in distance due to rotation around origin is

$\textcolor{g r e e n}{\overline{A B} - \overline{A ' B} = \sqrt{45} - \sqrt{17} \approx 2.59}$ units

#### Explanation:

Point A (5,2), Point B (2, -4)

Point A rotated by $\frac{3 \pi}{2}$ about the origin clockwise.

$A \left(\begin{matrix}5 \\ 2\end{matrix}\right) \to A ' \left(\begin{matrix}- 2 \\ 5\end{matrix}\right)$

Distance formula $d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$,

$\overline{A B} = \sqrt{{\left(5 - 2\right)}^{2} + {\left(2 - \left(- 4\right)\right)}^{2}} = \sqrt{45}$

$\overline{A ' B} = \sqrt{{\left(- 2 - 2\right)}^{2} + {\left(5 - \left(- 4\right)\right)}^{2}} = \sqrt{17}$

Reduction in distance due to rotation around origin is

$\textcolor{g r e e n}{\overline{A B} - \overline{A ' B} = \sqrt{45} - \sqrt{17} \approx 2.59}$ units