Question

A line segment has endpoints at `(3 ,2 )` and `(7 , 5)`. The line segment is dilated by a factor of `4` around `(2 , 3)`. What are the new endpoints and length of the line segment?

Answer 1

`(6,-1),(22,11),20" units"`

Explanation:

let `A=(3,2),B=(7,5)" and " C=(2,3)`

`"and " A',B'" be the image of A and B under the dilatation"`

`vec(CA)=ula-ulc=((3),(2))-((2),(3))=((1),(-1))`

`rArrvec(CA')=4((1),(-1))=((4),(-4))`

`rArrA'=(2+4,3-4)=(6,-1)color(red)(larr)`

`vec(CB)=ulb-ulc=((7),(5))-((2),(3))=((5),(2))`

`rArrvec(CB')=4((5),(2))=((20),(8))`

`rArrB'=(2+20,3+8)=(22,11)color(red)(larr)`

To calculate 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)|)))`

`"let " (x_1,y_1)=(6,-1)" and " (x_2,y_2)=(22,11)`

`d_(A'B')=sqrt((22-6)^2+(11+1)^2)`

`color(white)(d_(A'B'))=sqrt(256+144)=sqrt400`

`rArrd_(A'B')=20" units"`