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 $A'B' = sqrt((3-4)^2 + (8/3-13/3)^2)$

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

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