A line segment goes from #(2 ,6 )# to #(1 ,3 )#. The line segment is dilated about #(2 ,0 )# by a factor of #2#. Then the line segment is reflected across the lines #x = 2# and #y=5#, in that order. How far are the new endpoints from the origin?

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Answer 1 · 2018-04-12T04:04:22

#color(green)("Dist of A from origin after dilation and reflections " = 2.83#

#color(green)("Dist of B from origin after dilation and reflections " = 5.66#

#A(2,6), B(1,3), " Dilated about " C(2,0) " by factor " 3#

#A'(x,y) -> 2 * A(x,y) - C(x,y) = ((4,12) - (2,0)) = (2,12)#

#B'(x,y) -> 2 * B(x,y) - C(x,y) = ((2,6) - (2,0)) = (0,6)#

Line segment A'B' reflected across x = 2, y = 5 in that order.

Reflection Rule :

#color(crimson)("reflect over a line. ex: x=h. (2h-x, y)"#

#A'(x,y) -> A''(x,y) = (4 - 2, 12) -> (2,12)

#B'(x,y) - > B''(x,y) = (4-0, 6) -> (4,6)#

#color(crimson)("reflect over a line. ex: y= k. (x, 2k-y)"#

#A''(x,y) -> A'''(x,y) = (2, 10-12) -> (2,-2)#

#B''(x,y) -> B'''(x,y) = (4, 10-6) -> (4,4)#

#bar(A'''O) = sqrt(2^2 + 2^2) = 2.83#

#bar(B'''O) = sqrt(4^2 + 4^2) = 5.66#

#color(green)("Dist of A from origin after dilation and [reflections](https://socratic.org/geometry/transformations/reflections) " = 2.83#

#color(green)("Dist of B from origin after dilation and reflections " = 5.66#