Question

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?

Answer 1

A(2 ,-4), change ≈ 3.685

Explanation:

Before rotating point A, let's calculate the distance ( d) between A and 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 coordinate points"`

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

let `(x_1,y_1)=(4,2)" and " (x_2,y_2)=(3,1)`

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

Under a rotation, clockwise about origin of `pi/2`

a point (x ,y) → (y ,-x)

`rArrA(4,2)toA(2,-4)larr" 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