Point A is at #(-3 ,-4 )# and point B is at #(5 ,8 )#. 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?

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Answer 1 · 2018-03-31T16:56:47

#A'(3, 4)#

The distance has decreased from #4sqrt(13) to 2sqrt(5)#

Given: #A(-3, -4), B(5, 8)#; rotate #A# by #pi " radians"=180^@# clockwise (CW).

distance #AB = sqrt((8- -4)^2 + (5 - -3)^2) = sqrt(12^2 + 8^2)#

# AB= sqrt(208) = sqrt(16*13) = sqrt(16)sqrt(13) = 4sqrt(13)~~14.422#

A CW #pi = 180^@# transformation is #(x, y) -> (-x, -y)#

New transformation: #" "A' = (3, 4)#

distance #A'B = sqrt((8-4)^2 + (5-3)^2) = sqrt(4^2 + 2^2)#

#A'B = sqrt(20) = sqrt(4)sqrt(5) = 2sqrt(5) ~~4.47#

The distance has decreased from #4sqrt(13) to 2sqrt(5)#

The distance has decreased from #~~14.422 to ~~4.47# which is a decrease of #~~9.950#

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