A line segment has endpoints at #(5 ,5 )# and #(3 ,7 )#. The line segment is dilated by a factor of #1/2 # around #(3 , 1)#. What are the new endpoints and length of the line segment?

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Answer 1 · 2018-02-05T09:36:22

New end points #color(green)((4,3)# and #color(green)(3,4)#

Length of the line segment after dilation is #color(purple)(1.4142)#

Given : A (5,5), B (3,7), Dilation Point c(3,1), Dilation factor 1/2

To find new end points A', B' and A'B'

#a' - c = (1/2) (a - c)#

#a' = (1/2)a + (1/2)c#

#a' = (1/2)((5),(5)) + (1/2)((3),(1))= ((5/2),(5/2)) + ((3/2),(1/2)) = ((4),(3))#

#A' (4 , 3)#

#b' - c = (1/2) (b - c)#

#b' = (1/2)b + (1/2)c = (1/2)((3),(7)) + (1/2)((3),(1))#

#=> ((3/2),(7/2)) + ((3/2),(1/2)) = ((3),(4))#

#B' (3, 4)#

using distance formula between two points,

#bar(A'B') = sqrt((4-3)^2 + (3-4)^2) ~~ 1.4142#