Question

Point A is at `(-7 ,3 )` and point B is at `(5 ,4 )`. 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

`A' = (7, -3)`
The distance has decreased by `sqrt(145) - sqrt(53) ~~ 4.76`

Explanation:

Given: `A(-7, 3), B(5, 4)` Point `A` is rotated `pi` clockwise about the origin.

A `pi` rotation clockwise is a `180^@` rotation CW.

The coordinate rule for a `180^@` rotation CW is: `(x, y) -> (-x, -y)`

Using the coordinate rule: `A' = (7, -3)`

The distance formula is `d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

`d_(AB) = sqrt((5 - -7)^2 + (4 - 3)^2) = sqrt(12^2 + 1^2) = sqrt(145)`

`d_(A'B) = sqrt((5-7)^2 + (4 - -3)^2) = sqrt(2^2 + 7^2) = sqrt(53)`

The distance between point `A` and `B` has not changed.

The distance between the rotated point `A` = `A'` and `B` is `sqrt(145) - sqrt(53) ~~4.76`