Question

A line segment has endpoints at `(7 ,2 )` and `(1 ,8 )`. The line segment is dilated by a factor of `6` around `(4 ,5 )`. What are the new endpoints and length of the line segment?

Answer 1

`(22,-13),(-14,23),~~37.4`

Explanation:

`"label the endpoints "A(7,2)" and "B(1,8)`

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

`"let the centre of dilatation be C"`

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

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

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

`rArrula'=6ula-5ulc`

`color(white)(rArrula')=6((7),(2))-5((4),(5))`

`color(white)(rArrula')=((42),(12))-((20),(25))=((22),(-13))`

`rArrA'=(22,-13)`

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

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

`rArrulb'=6ulb-5ulc`

`color(white)(rArrulb')=6((1),(8))-5((4),(5))`

`color(white)(rArrulb')=((6),(48))-((20),(25))=((-14),(23))`

`rArrB'=(-14,23)`

`"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)=(22,-13)" and "(x_2,y_2)=(-14,23)`

`d=sqrt((-14-22)^2+(23-13)^2)=sqrt(1296+100)`

`rArrd=sqrt1396~~37.4" to 1 dec. place"`