Question

Point A is at `(8 ,-4 )` and point B is at `(2 ,-3 )`. Point A is rotated `(3pi)/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(8,-4)to(4,8)," change"≈5.097`

Explanation:

Under a clockwise rotation about the origin of `(3pi)/2`

a point (x ,y) → (-y ,x)

Hence A (8 ,-4) → A(4 ,8)

To calculate the distance (d) between A and B use the `color(blue)"distance formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)color(white)(2/2)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"`

The 2 points here are A(8 ,-4) and B(2 ,-3). That is the original A and B.

let `(x_1,y_1)=(8,-4)" and " (x_2,y_2)=(2,-3)`

`d=sqrt((2-8)^2+(-3+4)^2)=sqrt37≈6.083`

Now use the 'new' point A(4 ,8) and B(2 ,-3)

`d=sqrt((2-4)^2+(-3-8)^2)=sqrt125≈11.180`

change in distance between A and B = 11.180 - 6.083 = 5.097