A line segment has endpoints at #(3 ,8 )# and #(5 , 9)#. The line segment is dilated by a factor of #3 # around #(2 , 3)#. What are the new endpoints and length of the line segment?

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Answer 1 · 2018-05-23T19:41:56

#color(blue)(A'=(5,18)#
#color(blue)(B'=(11,21)#
#color(blue)(d=3sqrt(5))#

We can preform a dilation using vectors.

Let #A=(3,8)#, #B=(5,9)#, #D=(2,3)# and #O# be the origin.

#vec(OD)=((2),(3))#

#vec(DA)=((1),(5))#

#vec(DB)=((3),(6))#

Dilating A:

#vec(OA)=vec(OD)+3vec(DA)=((2),(3))+3((1),(5))=((5),(18))#

Dilating B:

#vec(OB)=vec(OD)+3vec(DB)=((2),(3))+3((3),(6))=((11),(21))#

New endpoints:

#A'=(5,18)# and #B'=(11,21)#

Length of line segment is found using the distance formula:

#d=sqrt((11-5)^2+(21-18)^2)=sqrt(45)=color(blue)(3sqrt(5))#