# Point A is at (3 ,2 ) and point B is at (-7 ,3 ). Point A is rotated pi/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?

Mar 21, 2016

Rotating Point A $\left(3 , 2\right)$ clockwise through $\frac{\pi}{2}$ yields Point A' $\left(2 , - 3\right)$. We can then calculate the distance between Point A' and Point B as approximately $10.8$ $u n i t s$.

#### Explanation:

Now, $\frac{\pi}{2}$ is $\frac{1}{4}$ of a full rotation ($2 \pi$) or ${90}^{o}$. If we rotate the point $\left(3 , 2\right)$ through this angle clockwise, we move from the first to the fourth quadrant, so the $x$ value will still be positive and the $y$ value will be negative.

You should probably draw a diagram, but it turns out that the new point, which we can call Point A', has the coordinates $\left(2 , - 3\right)$.

We can calculate the distance between Point A' and Point B:

$r = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}} = \sqrt{{\left(\left(- 7\right) - 2\right)}^{2} + {\left(3 - \left(- 3\right)\right)}^{2}} = \sqrt{81 + 36} \approx 10.8$