Point A is at #(4 ,2 )# and point B is at #(3 ,1 )#. Point A is rotated #pi/2 # 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
Sep 27, 2016
A(2 ,-4), change ≈ 3.685
Explanation:
Before rotating point A, let's calculate the distance ( d) between A and 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"# The 2 points here are A(4 ,2) and B(3 ,1)
let
# (x_1,y_1)=(4,2)" and " (x_2,y_2)=(3,1)#
#d_1=sqrt((3-4)^2+(1-2)^2)=sqrt(1+1)=sqrt2≈1.414# Under a rotation, clockwise about origin of
#pi/2# a point (x ,y) → (y ,-x)
#rArrA(4,2)toA(2,-4)larr" new coordinates of point A"# Calculate the distance between A(2 ,-4) and B(3 ,1)
#d_2=sqrt((3-2)^2+(1+4)^2)=sqrt(1+25)=sqrt26≈5.099# change in distance between A and B = 5.099 - 1.414 = 3.685
