Question

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

Answer 1

New Endpoints are `(-2, -1)` and `(6, -5)`
Length `l=8.94427" "`

Explanation:

Let `A(x_a, y_a)=(1, 2)`
Let `B(x_b, y_b)=(3, 1)`
Let `R(x_r, y_r)=(2, 3)`

Let the new endpoints be
`C(x_c, y_c)` and `D(x_d, y_d)`

Let `k=4`

Solve the new endpoints by ratio and proportion

`(RC)/(RA)=k`

`(x_c-x_r)/(x_a-x_r)=4`
`(x_c-2)/(1-2)=4`

`x_c=-2`

`(y_c-y_r)/(y_a-y_r)=4`
`(y_c-3)/(2-3)=4`

`y_c=-1`

`(RD)/(RB)=k`
`(x_d-x_r)/(x_b-x_r)=4`
`(x_d-2)/(3-2)=4`
`x_d=6`

`(y_d-y_r)/(y_b-y_r)=4`
`(y_d-3)/(1-3)=4`
`y_d=-5`

Length

`l=sqrt((x_d-x_c)^2+(y_d-y_c)^2)`
`l=4sqrt(5)=8.94427" "`

God bless....I hope the explanation is useful