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

1 Answer
Apr 21, 2018

I get # 4 sqrt{2}# for the first endpoint and #0# for the second.

Explanation:

Yikes, that's a lot of transformations. We're just interested in the image of each endpoint.

To do the dilation we start by getting a direction vector from the dilation point to each endpoint, essentially translating the dilation point to the origin.

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

We dilate each direction vector by a factor of two and translate back:

# (2,2)+2(2,-1)=(6,0) quad quad quad quad (2,2)+2(0,1)=(2,4)#

Reflecting through #x=-2# leaves the #y# coordinate alone:

# (6,0) to (2-6,0)=(-4,0) quad quad quad quad (2,4) to (2-2,4)=(0,4)#

Reflecting through #y=4# leaves the #x# coordinate alone:

# (-4,0) to (-4,4-0)=(-4,4) quad quad quad quad (0,4) to(0,0) #

If I did that right the first endpoint is #\sqrt{4^2+4^2}=4\sqrt{2}# from the origin and the second endpoint is the origin, so a distance of zero.