A line segment has endpoints at #(5 ,5 )# and #(3 ,5 )#. The line segment is dilated by a factor of #1/2 # around #(3 , 2)#. What are the new endpoints and length of the line segment?

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Answer 1 · 2017-06-28T10:14:14

The new endpoints are #(4,7/2)# and #(3,7/2)# and the length of the line segment is #=1#

Let the line segment be #AB#

#A=(5,5)# and

#B=(3,5)#

#C=(3,2)#

The length of the segment #AB=2#

Let point #A'=(x,y)# be the point after dilatation, then

#vec(CA')=1/2 vec(CA)#

#((x-3),(y-2))=1/2((5-3),(5-2))=1/2((2),(3))#

Therefore,

#x-3=1#, #=>#, #x=4#

#y-2=3/2#, #=>#, #y=7/2#

The point #A'=(4,7/2)#

Let point #B'=(p,q)# be the point after dilatation, then

#vec(CB')=1/2 vec(CB)#

#((p-3),(q-2))=1/2((3-3),(5-2))=1/2((0),(3))#

Therefore,

#p-3=0#, #=>#, #p=3#

#q-2=3/2#, #=>#, #q=7/2#

The point #B'=(3,7/2)#

The length of the segment #A'B'=1#