A line segment has endpoints at #(5 ,6 )# and #(2 , 1)#. The line segment is dilated by a factor of #3 # around #(6 , 2)#. What are the new endpoints and length of the line segment?

Dilation is a fancy word for scale. As soon as you talk about scale your are dealing with ratios. Dilation that is positive increases the magnitude. Assuming positive dilation as not indicated otherwise.

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#color(blue)("Determine point B' ")" "larr# B prime

#(OB)/(OB') = 1/3#

#(x_O-x_B)/ (x_O-x_B') =1/3#

#=>3(x_O-x_B)=x_O-x_B'#

#=>3(6-5)=6-x_B'#

#x_B'=3#
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#(y_O-y_B)/ (y_O-y_B') =1/3#

#=>3(y_O-y_B)=y_O-y_B'#

#=>3(2-6)=2-y_B'#

#y_B'=14#

#color(blue)("Point B'"->(x,y)=(3,14))#
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#color(blue)("Determine point A' ")" "larr# A prime

#(OA)/(OA') = 1/3#

#(x_O-x_A)/ (x_O-x_A') =1/3#

#=>3(x_O-x_A)=x_O-x_A'#

#=>3(6-2)=6-x_A'#

#x_A'=-6#
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#(y_O-y_A)/ (y_O-y_A') =1/3#

#=>3(y_O-y_A)=y_O-y_A'#

#=>3(2-1)=2-y_A'#

#y_A'=-1#

#color(blue)("Point A'"->(x,y)=(-6,-1))#
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#color(blue)("Determine length of A'B'")#

#L=sqrt((x_B'-x_A')^2+(y_B'-y_B')^2)#

#color(blue)(L=sqrt(9^2+15^2) = sqrt(306))#
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#color(blue)("Check")#

#(AB)/(A'B') = 1/3#

Length #AB = sqrt((5-2)^2+(6-1)^2) =sqrt( 34)#

#LHS ->RHS#

#1/3->sqrt(34)/(sqrt(306))=1/3#

#LHS=RHS color(red)(larr" True")#