A line segment has endpoints at #(5 ,9 )# and #(6 ,7 )#. The line segment is dilated by a factor of #2 # around #(5 ,3 )#. What are the new endpoints and length of the line segment?

1 Answer
Jim G.
Jul 27, 2018

#(5,15),(7,11),2sqrt5#

Explanation:

#"let "A=(5,9),B=(6,7)" and "C=(5,3)" then"#

#vec(CA')=2vec(CA)" where A' is the image of A"#

#ula'-ulc=2ula-2ulc#

#ula'=2ula-ulc#

#color(white)(ula')=2((5),(9))-((5),(3))#

#color(white)(ula')=((10),(18))-((5),(3))=((5),(15))#

#rArrA'=(5,15)#

#"Similarly "#

#ulb'=2ulb-ulc#

#color(white)(ulb')=2((6),(7))-((5),(3))#

#color(white)(ulb')=((12),(14))-((5),(3))=((7),(11))#

#rArrB'=(7,11)#

#"calculate the length using the "color(blue)"distance formula"#

#•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#"let "(x_1,y_1)=(7,11)" and "(x_2,y_2)=(5,15)#

#d=sqrt((5-7)^2+(15-11)^2#

#color(white)(d)=sqrt(4+16)=sqrt20=2sqrt5#

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