# Point A is at (-3 ,-4 ) and point B is at (5 ,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?

Mar 31, 2018

$A ' \left(3 , 4\right)$

The distance has decreased from $4 \sqrt{13} \to 2 \sqrt{5}$

#### Explanation:

Given: $A \left(- 3 , - 4\right) , B \left(5 , 8\right)$; rotate $A$ by $\pi \text{ radians} = {180}^{\circ}$ clockwise (CW).

distance $A B = \sqrt{{\left(8 - - 4\right)}^{2} + {\left(5 - - 3\right)}^{2}} = \sqrt{{12}^{2} + {8}^{2}}$

$A B = \sqrt{208} = \sqrt{16 \cdot 13} = \sqrt{16} \sqrt{13} = 4 \sqrt{13} \approx 14.422$

A CW $\pi = {180}^{\circ}$ transformation is $\left(x , y\right) \to \left(- x , - y\right)$

New transformation: $\text{ } A ' = \left(3 , 4\right)$

distance $A ' B = \sqrt{{\left(8 - 4\right)}^{2} + {\left(5 - 3\right)}^{2}} = \sqrt{{4}^{2} + {2}^{2}}$

$A ' B = \sqrt{20} = \sqrt{4} \sqrt{5} = 2 \sqrt{5} \approx 4.47$

The distance has decreased from $4 \sqrt{13} \to 2 \sqrt{5}$

The distance has decreased from $\approx 14.422 \to \approx 4.47$ which is a decrease of $\approx 9.950$