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,

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

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