A line segment has endpoints at #(7 ,6 )# and #(5 ,8 )#. The line segment is dilated by a factor of #4 # around #(2 ,1 )#. What are the new endpoints and length of the line segment?
1 Answer
Explanation:
Let the endpoints be A (7 ,6) and B (5 ,8) and their images be A' and B', respectively, under the dilation.
Let the centre of dilatation be C (2 ,1)
#vec(CA)=ula-ulc=((7),(6))-((2),(1))=((5),(5)) #
#rArrvec(CA')=4((5),(5))=((20),(20))#
#rArrA'=(2+20,1+20)=(color(red)(22,21))# Similar process to obtain coordinates of B'
#vec(CB)=ulb-ulc=((5),(8))-((2),(1))=((3),(7))#
#rArrvec(CB')=4((3),(7))=((12),(28))#
#rArrB'=(2+12,1+28)=(color(red)(14,29))#
#"new endpoints are "(22,21)" and " (14,29)# To calculate the length, use the
#color(blue)"distance formula"#
#color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))#
where# (x_1,y_1),(x_2,y_2)" are 2 coordinate points"# The 2 points here are (22 ,21) and (14 ,29)
let
# (x_1,y_1)=(22,21)" and " (x_2,y_2)=(14,29)#
#d=sqrt((14-22)^2+(29-21)^2)=sqrt(64+64)=sqrt128#
#"length of line segment" =sqrt128=8sqrt2≈11.31#
