A line segment goes from #(1 ,2 )# to #(4 ,7 )#. The line segment is reflected across #x=6#, reflected across #y=1#, and then dilated about #(1 ,1 )# by a factor of #2#. How far are the new endpoints from the origin?
1 Answer
Original segment
is transformed into
The distances from the origin to the new endpoints are
Explanation:

Reflection of a point with coordinates
#(a_0,b_0)# relative to a line#x=6# (vertical line intersecting Xaxis at coordinate#x=6# ) will be horizontally shifted into a new Xcoordinate obtained by adding to an Xcoordinate of the axis of symmetry (#x=6# ) the distance from it of the original Xcoordinates (#6a_0# ).
Ycoordinate remains the same in this transformation.
So, new coordinates are:
#(a_1,b_1) = (6+(6a_0),b_0)=(12a_0,b_0)# 
Reflection of a point with coordinates
#(a_1,b_1)# relative to a line#y=1# (horizontal line intersecting Yaxis at coordinate#y=1# ) will be vertically shifted into a new Ycoordinate obtained by adding to an Ycoordinate of the axis of symmetry (#y=1# ) the distance from it of the original Ycoordinates (#1b_1# ).
Xcoordinate remains the same in this transformation.
So, new coordinates are:
#(a_2,b_2) = (a_1,1+(1b_1))=#
# = (a_1,2b_1)=(12a_0,2b_0)# 
Dilation about a center point
#(1,1)# by a factor of#2# will transform a point#(a_2,b_2)# into
#(a_3,b_3) = (1+2(a_21),1+2(b_21)) =#
# = (1+2(12a_01),1+2(2b_01)) =#
# = (232a_0, 52b_0)# 
Using this formula for both ends of our original segment
#AB# , where#A(1,2)# and#B=(4,7)# :
4.1.#(a_0=1, b_0=2)#
# rarr# #(a_3=232*1, b_3=52*2) = #
# = (21, 9)#
4.2.#(a_0=4, b_0=7)#
# rarr# #(a_3=232*4, b_3=52*7) = #
# = (15, 19)# 
The distance of each end of a new segment from the origin are
#d_A = sqrt((21)^2+(9)^2) = sqrt(552) ~~22.8 #
#d_B = sqrt((15)^2+(19)^2) = sqrt(586) ~~24.2 #