Question

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

Answer 1

New endpoints: Dilation of `(2,1) rarr (-1,1)`
`color(white)("New endpoints:")`Dilation of `(4,2) rarr (7,5)`
New line segment length: `4sqrt(5)" units"`

Explanation:

If the center of dilation is `C=(3,1)`
and original endpoints are
#{: (P=(2,1),color(white)("xxxxx"),Q=(4,2)), (vec(CP)=(2-3,1-1),,vec(CQ)=(4-3,2-1)), (color(white)(vec(CP))=(-1,0),,color(white)(vec(CQ))=(1,1)), ("dilated vector:",,"dilated vector:"), (4vec(CP)=(-4,0),,4vec(CQ)=(4,4)), (hat(P)=C+4vec(CP),,hat(Q)=C+4vec(CQ)), (color(white)(hat(P))=(-1,1),,color(white)(hat(Q))=(7,5)) :}#

The length of the dilated line segment is
`abs(hatPhatQ)=sqrt((7-(-1))^2+(5-1)^2)=sqrt(80)=4sqrt(5)`