Question

Point A is at `(3 ,-2 )` and point B is at `(2 ,1 )`. 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

≈ 1.937

Explanation:

Under a rotation of `pi " about the origin "`

a point (x , y) → (-x , -y)

hence A(3 , -2) → (-3 , 2)

Now , we have to calculate the difference between 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 coordinate points "`

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

`d_(AB) = sqrt((2-3)^2 + (1+2)^2) = sqrt(1+9) = sqrt10 ≈ 3.162`

For length A'B, let `(x_1,y_1)=(-3,2)" and " (x_2,y_2)=(2,1)`

`d_(A'B) = sqrt((2+3)^2 + (1-2)^2) = sqrt(25+1)=sqrt26 ≈ 5.099`

difference = A'B - AB = 5.099 - 3.162 = 1.937