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?
1 Answer
Apr 14, 2016
≈ 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
