# Point A is at (4 ,2 ) and point B is at (3 ,1 ). 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?

Sep 27, 2016

A(2 ,-4), change ≈ 3.685

#### Explanation:

Before rotating point A, let's calculate the distance ( d) between A and B, using the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

The 2 points here are A(4 ,2) and B(3 ,1)

let $\left({x}_{1} , {y}_{1}\right) = \left(4 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(3 , 1\right)$

d_1=sqrt((3-4)^2+(1-2)^2)=sqrt(1+1)=sqrt2≈1.414

Under a rotation, clockwise about origin of $\frac{\pi}{2}$

a point (x ,y) → (y ,-x)

$\Rightarrow A \left(4 , 2\right) \to A \left(2 , - 4\right) \leftarrow \text{ new coordinates of point A}$

Calculate the distance between A(2 ,-4) and B(3 ,1)

d_2=sqrt((3-2)^2+(1+4)^2)=sqrt(1+25)=sqrt26≈5.099

change in distance between A and B = 5.099 - 1.414 = 3.685