Question

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?

Answer 1

(4 , -5), difference ≈ 11.65

Explanation:

Under a rotation of `pi " 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 `color(blue)" 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 `(x_1,y_1)" and " (x_2,y_2)" are 2 points "`

calculate AB

let `(x_1,y_1)=(-4 ,5)" and " (x_2,y_2)=(-3 ,7)`

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

calculate A'B
let `(x_1,y_1)=(4 ,-5)" and " (x_2,y_2)=(-3 ,7)`

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

difference in length = 13.89 - 2.24 = 11.65