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A line segment has endpoints at #(2 ,1 )# and #(6 ,2 )#. The line segment is dilated by a factor of #4 # around #(2 ,1 )#. What are the new endpoints and length of the line segment?

Answer 1

Maverick Smith

New end points coordinates of #A ((2),(1)), B ((18),(5))#

length of line segment #~~ color(green)(16.49#

Given : A(2,1), B(6,2), dilated around C(2,1), dilation factor 4

To find the end points of the line segment and its length

#vec(A'C) = 4 * vec(AC)# or #a' = 4a - 3c#

#a' = 4((2),(1)) - 3 ((2),(1)) = ((8),(4)) - ((6),(3)) = color(brown)(((2),(1))#

#vec(B'C) = 4 * vec(BC)# or #b' = 4b - 3c#

#b' = 4((6),(2)) - 3 ((2),(1)) = ((24),(8)) - ((6),(3)) = color(brown)( ((18),(5))#

Length of the line segment using distance formula,

#vec(A'B') = sqrt((2-18)^2 + (1-5)^2) ~~ color(green)(16.49#

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Answer 2

Ella Armstrong

The new endpoints of the dilated line segment are (2, 1) and (10, 5). The length of the original line segment is ( \sqrt{(6-2)^2 + (2-1)^2} = \sqrt{16 + 1} = \sqrt{17} ), and after dilation, the length becomes ( 4 \times \sqrt{17} ), which is approximately 16.49 units.

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