Question

Point A is at `(9 ,-2 )` and point B is at `(1 ,-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?

Answer 1

(-2 ,-9) , ≈ 1.13

Explanation:

Under a rotation of `pi/2" clockwise about 0 "`

a point (x ,y) → (y ,-x)

hence A (9 ,-2) → A' (-2 , -9)

To calculate the change in distance we need to find the length 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))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 points "`

calculate AB

let `(x_1,y_1)=(9,-2)" and " (x_2,y_2)=(1,-3)`

`d=sqrt((1-9)^2+(-3+2)^2)=sqrt(64+1)=sqrt65 ≈ 8.06`

calculate A'B

let `(x_1,y_1)=(-2,-9)" and " (x_2,y_2)=(1,-3)`

`d=sqrt((1+2)^2+(-3+9)^2)=sqrt(9+36)=sqrt45 ≈ 6.93`

change in length = 8.06 - 6.93 = 1.13