A line segment goes from #(1 ,1 )# to #(4 ,2 )#. The line segment is reflected across #x=2#, reflected across #y=-1#, and then dilated about #(1 ,1 )# by a factor of #2#. How far are the new endpoints from the origin?

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Answer 1 · 2018-04-12T04:44:31

#color(purple)("Distance of A from origin after reflection and dilation " = 8.6#

#color(purple)("Distance of B from origin after reflection and dilation " = 9.06#

#A(1,1), B(4,2) " reflected across " x = 2, y = -1 " in that order"#

#"Reflection rule : reflect thru " x = 2, y = -1, h=2, k= -1. (2h-x, 2k-y)#

#A'(x,y) = (2h-x, 2k-y) = (4-1, -2-1) = (3, -3)#

#B'(x,y) = (2h-x, 2k-y) = (4-4, -2-2) = (0, -4)#

#"A', B' dilated about C (1,1) by a factor of 2"#

#A'(x,y) -> A''(x,y) = 2*A'(x,) - C(x,y) = ((6,-6)-(1,1)) = (5,-7)#

#B'(x,y) -> B''(x,y) = 2*B'(x,y) - C(x,y) = ((0,-8)-(1,1)) = (-1,-9)#

#OA'' = sqrt(5^2 + 7^2) = 8.6#

#OB'' = sqrt(15^2 + 9^2) = 9.06#

#color(purple)("Distance of A from origin after reflection and dilation " = 8.6#

#color(purple)("Distance of B from origin after reflection and dilation " = 9.06#