A line segment goes from #(3 ,4 )# to #(5 ,1 )#. The line segment is dilated about #(1 ,0 )# by a factor of #2#. Then the line segment is reflected across the lines #x=-2# and #y=2#, in that order. How far are the new endpoints from the origin?

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Answer 1 · 2018-04-21T11:39:19

If I've done it right the first is #sqrt{85}# from the origin and the second is #11# from the origin.

Thought I just did one of these.

Shifting the dilation point to the origin maps the other points:

#(3,4)-(1,0)=(2,4) quad quad quad quad (5,1)-(1,0)=(4,1)#

Doubling each dilates around the origin by a factor of two.

# 2(2,4) = (4,8) quad quad quad quad 2(4,1)=(8,2)#

Now we shift the origin back to the dilation point:

# (4,8)+(1,0)=(5,8) quad quad quad quad (8,2)+(1,0)=(9,2)#

Now we reflect through #x=-2# which leaves the #y# coordinate alone.

# (-2 - 5,8)=(-7,8) quad quad quad quad (-2-9,2)=(-11,2)#

Now we reflect through #y=2# which leaves the #x# coordinate alone.

# (-7,2-8)=(-7,-6) quad quad quad quad (-11,2-2)=(-11,0)#

If I've done that right the first is #sqrt{7^2+6^2}=sqrt{85}# from the origin and the second is #11# from the origin.