A line segment has endpoints at #(1 ,4 )# and #(4 , 9)#. The line segment is dilated by a factor of #1/3 # around #(4 , 2)#. What are the new endpoints and length of the line segment?

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Answer 1 · 2018-06-07T11:10:47

#color(brown)("Length of the line segment " color(blue)(bar(AB) " after dilation is " color(green)(1.94 " units"#

#A (1,4), B(4,9). " Dilated by a factor of 1/3 around " C (4,2)#

#bar(A'C) = (1/3) bar(AC)#

#A'C - c = (1/3)(a - c)#

#A' = a / 3 + (2/3)c#

#A'((x),(y)) = (1/3)((1),(4)) + (2/3)((4),(2)) = ((3),(8/3))#

Similarly, #B' = b / 3 + (2/3)c#

#B'((x),(y)) = (1/3)((4),(9)) + (2/3)((4),(2)) = ((4),(13/3))#

Length of the line segment #A'B' = sqrt((3-4)^2 + (8/3-13/3)^2)#

#vec(A'B') = 1.94 ' units"#