# Point A is at (-4 ,5 ) and point B is at (-3 ,7 ). 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?

Apr 25, 2016

(4 , -5), difference ≈ 11.65

#### Explanation:

Under a rotation of $\pi \text{ about the origin }$

a point (x , y) → (-x ,-y)

hence A (-4 ,5) → A' (4 ,-5)

To find the change in distance we require to calculate the lengths of AB and A'B using the $\textcolor{b l u e}{\text{ distance formula }}$

color(red)(|bar(ul(color(white)(a/a)color(black)( d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2))|))
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points }$

calculate AB

let $\left({x}_{1} , {y}_{1}\right) = \left(- 4 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 3 , 7\right)$

 d =sqrt((-3+4)^2+(7-5)^2)=sqrt(1+4)=sqrt5 ≈ 2.24

calculate A'B
let $\left({x}_{1} , {y}_{1}\right) = \left(4 , - 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 3 , 7\right)$

 d=sqrt((-3-4)^2+(7+5)^2)=sqrt(49+144)=sqrt193 ≈ 13.89

difference in length = 13.89 - 2.24 = 11.65