Question

A line segment has endpoints at `(1 ,2 )` and `(3 ,4 )`. The line segment is dilated by a factor of `6` around `(2 ,5 )`. What are the new endpoints and length of the line segment?

Answer 1

`(-4,-13),(8,-1),~~16.97`

Explanation:

`"label the endpoints "A(1,2)" and "B(3,4)`

`"label the centre of dilatation "C(2,5)`

`"let A' and B' be the images of A and B"`

`"then"`

`vec(CA')=color(red)(6)vec(CA)`

`rArrula'-ulc=6(ula-ulc)`

`rArrula'-ulc=6ula-6ulc`

`rArrula'=6ula-5ulc`

`color(white)(rArrula')=6((1),(2))-5((2),(5))`

`color(white)(rArrula')=((6),(12))-((10),(25))=((-4),(-13))`

`rArrA'=(-4,-13)`

`"and "`

`vec(CB')=color(red)(6)vec(CB)`

`rArrulb'-ulc=6(ulb-ulc)`

`rArrulb'=6ulb-5ulc`

`color(white)(rArrulb')=6((3),(4))-5((2),(5))`

`color(white)(rArrulb)=((18),(24))-((10),(25))=((8),(-1))`

`rArrB'=(8,-1)`

`"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)=(-4,-13)" and "(x_2,y_2)=(8,-1)`

`d=sqrt((8+4)^2+(-1+13)^2)`

`color(white)(d)=sqrt(144+144)=sqrt288=12sqrt2~~16.97" 2 d.p"`