Point A is at #(-1 ,-8 )# and point B is at #(-5 ,3 )#. 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
May 21, 2016
(-8 ,1) and ≈ 8.104
Explanation:
Under a rotation of
#pi/2# clockwise about O.a point (x ,y) → (y ,-x)
hence point A(-1 ,-8) → A' (-8 ,1)
To find the change in distance between AB and A'B we require to calculate the length of AB and A'B.
We can do this 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 points"# length of AB
# (x_1,y_1)=(-1 ,-8)" and " (x_2,y_2)=(-5,3)#
#d_(AB)=sqrt((-5+1)^2+(3+8)^2)=sqrt(16+121)≈11.705# length of A'B
#(x_1,y_1)=(-8 ,1)" and " (x_2,y_2)=(-5, 3)#
#d_(A'B)=sqrt((-5+8)^2+(3-1)^2)=sqrt(9+4)≈3.601# hence the change in length = 11.705 - 3.601 = 8.104
