Question

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

The distance has changed by `abs(bar(AB) - bar(A^’B)) = abs(sqrt(13)-sqrt(45))`

Explanation:

Consider the point `A(9,-2)`, a rotation by `pi` will put A at `A^’(9,2 )`. Point B is at `B(2,4)` we need to calculate the distance of `bar(AB)` and `bar(A^’B)` and compare the difference:
Distance Formula: `d = sqrt((x_2-x_1)^2 +(y_2-y_1)^2)`
`bar(AB) = sqrt((9-2)^2 +(-2-4)^2) = 49-36`
`bar(AB) = sqrt (13)`
`bar(A^’B) = sqrt((9-2)^2 +(2-4)^2)=49-4`
`bar(AB) = sqrt (45)`
The length of `bar(AB)` changed by `abs(sqrt(13)-sqrt(45))`