A triangle has corners at #(1, 2 )#, #( 2, 3 )#, and #( 1 , 5 )#. If the triangle is dilated by # 4 x# around #(3, 1)#, what will the new coordinates of its corners be?

Let the corners of the triangle be

#A=(1,2)#

#B=(2,3)#

#C=(1,5)#

And the point #D=(3,1)#

Let the corners of the triangle be

#A'=(x,y)#, #B'=(x',y')# and #C'=(x'',y'')# after dilatation.

#vec(DA')=4vec(DA)#

#((x-3),(y-1))=4((1-3),(2-1))=((-8),(4))#

#x-3=-8#, #=>#, #x=-5#

#y-1=4#, #=>#, #y=5#

So,

#A'=(-5,5)#

#vec(DB')=4vec(DB)#

#((x'-3),(y'-1))=4((2-3),(3-1))=((-4),(8))#

#x'-3=-4#, #=>#, #x'=-1#

#y'-1=8#, #=>#, #y'=9#

#B'=(-1,9)#

#vec(DC')=4vec(DC)#

#((x''-3),(y''-1))=4((1-3),(5-1))=((-8),(16))#

#x''-3=-8#, #=>#, #x''=-5#

#y''-1=16#, #=>#, #y''=17#

#C'=(-5,17)#

#"let the vertices of the triangle be"#

#A(1,2),B(2,3)" and " C(1,5)#

#"and A',B',C' be the images of A,B and C respectively"#
#"under the dilatation"#

#"let the centre of dilatation be "D(color(magenta)(3),color(blue)(1))#

#• vec(DA)=ula-uld=((1),(2))-((3),(1))=((-2),(1))#

#rArrvec(DA')=color(red)(4)vec(DA)=color(red)(4)((-2),(1))=((-8),(4))#

#rArrA'=(color(magenta)(3)-8,color(blue)(1)+4)=(-5,5)#

#• vec(DB)=ulb-uld=((2),(3))-((3),(1))=((-1),(2))#

#rArrvec(DB')=color(red)(4)vec(DB)=color(red)(4)((-1),(2))=((-4),(8))#

#rArrB'=(color(magenta)(3)-4,color(blue)(1)+8)=(-1,9)#

#• vec(DC)=ulc-uld=((1),(5))-((3),(1))=((-2),(4))#

#rArrvec(DC')=color(red)(4)vec(DC)=color(red)(4)((-2),(4))=((-8),(16))#

#rArrC'=(color(magenta)(3)-8,color(blue)(1)+16))=(-5,17)#