A line segment goes from #(3 ,2 )# to #(1 ,3 )#. The line segment is dilated about #(1 ,1 )# by a factor of #2#. Then the line segment is reflected across the lines #x=4# and #y=-3#, in that order. How far are the new endpoints from the origin?

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Answer 1 · 2018-04-12T07:58:36

#color(blue)("Distances of A & B from origin after dilation and reflection " 9.49, 13.04 " respectively"#

#A (3,2), B (1,3), " dilated by 2 about C(1,1)"#

#A(x,y) -> A'(x,y) = 2 * A(x,y) - C(x,y) = (2*(3,2)-(1,1)) = (5, 3)#

#B(x,y) -> B'(x,y) = 2 * B(x,y) - C(x,y) = (2*(1,3)-(1,1)) = (1, 5)#

Line segment A'B' reflected across x = 4, y = -3, in that order.

#color(crimson)("reflect thru x = 4, y = -3 ", h=4, k= -3. (2h-x, 2k-y)"#

#A''(x, y) = (2h - x, 2k-y) = (2*4 - 5, 2*-3 - 3) = (3, -9)#

#B''(x, y) = (2h - x, 2k-y) = (2*4 - 1, 2*-3 - 5) = (7, -11)#

#OA'' = sqrt(3^2 + 9^2) = 9.49#

#OB'' = sqrt(7^2 + 11^2) = 13.04#