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)`