# Point A is at (1 ,-4 ) and point B is at (-9 ,-8 ). 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 7, 2018

A moved from point $\left(\begin{matrix}1 \\ - 4\end{matrix}\right) \to \left(\begin{matrix}4 \\ 1\end{matrix}\right)$

Distance between A & B changed by $\approx \textcolor{b l u e}{+ 15.81}$

#### Explanation:

A (1,-4), B (-9,-8). Point A rotated clockwise about the origin by $\frac{3 \pi}{2}$ from IV quadrant to I quadrant

$A \left(\begin{matrix}1 \\ - 4\end{matrix}\right) \to A ' \left(\begin{matrix}4 \\ 1\end{matrix}\right)$

Using distance formula $\sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$\vec{A B} = \sqrt{{\left(1 + 9\right)}^{2} + {\left(- 4 + 8\right)}^{2}} = \sqrt{116} \approx \textcolor{b l u e}{10.77}$

$\vec{A ' B} = \sqrt{{\left(4 + 9\right)}^{2} + {\left(1 + 8\right)}^{2}} = \sqrt{250} \approx \textcolor{b l u e}{15.81}$