Question

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

Answer 1

`(22,21),(14,29)" and "8sqrt2`

Explanation:

Let the endpoints be A (7 ,6) and B (5 ,8) and their images be A' and B', respectively, under the dilation.
Let the centre of dilatation be C (2 ,1)

`vec(CA)=ula-ulc=((7),(6))-((2),(1))=((5),(5))`

`rArrvec(CA')=4((5),(5))=((20),(20))`

`rArrA'=(2+20,1+20)=(color(red)(22,21))`

Similar process to obtain coordinates of B'

`vec(CB)=ulb-ulc=((5),(8))-((2),(1))=((3),(7))`

`rArrvec(CB')=4((3),(7))=((12),(28))`

`rArrB'=(2+12,1+28)=(color(red)(14,29))`

`"new endpoints are "(22,21)" and " (14,29)`

To calculate the 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)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

The 2 points here are (22 ,21) and (14 ,29)

let `(x_1,y_1)=(22,21)" and " (x_2,y_2)=(14,29)`

`d=sqrt((14-22)^2+(29-21)^2)=sqrt(64+64)=sqrt128`

`"length of line segment" =sqrt128=8sqrt2≈11.31`