Question

A line segment has endpoints at `(2 ,4 )` and `(5 ,3 )`. The line segment is dilated by a factor of `3` around `(3 ,8 )`. What are the new endpoints and length of the line segment?

Answer 1

New end points: `hat(P): (0,-4)` and `hat(Q):(12,5)`
New segment length: `abs(hat(P)hat(Q))=15`

Explanation:

Let `P` be the original point `(2,4)`
Let `Q` be the original point `(5,3)`
and
Let `C` be the center of dilation, `(3,8)`

Consider the vector `vec(CP)`
`color(white)("XXX")vec(CP)=(2,4)-(3,8)=(-1,-4)`
Dilation by a factor of `3` will scale this vector up by a factor of `3`
So `P` will move to the new location:
`color(white)("XXX")hat(P)=C+3vec(CP)`
`color(white)("XXXX")=(3,8)+3(-1,4)`
`color(white)("XXXX")=(3-3,8-12)`
`color(white)("XXXX")=(0,-4)`

Similarly
`color(white)("XXX")vec(CQ)=(5,3)-(2,4)=(3,-1)`
and new location for `Q` at
`color(white)("XXX")hat(Q)=(3,8)+3(3,-1)`
`color(white)("XXXX")=(12,5)`

The length of the new line segment will be (using the Pythagorean Theorem)
`color(white)("XXX")abs(hat(P)hat(Q))=sqrt((12-0)^2+(5-(-4))^2)`
`color(white)("XXX")=sqrt(12^2+9^2)`
`color(white)("XXX")=sqrt(225)`
`color(white)("XXX")=15`