Question

How do you find the coordinates of the other endpoint of a segment with the given Endpoint: (1,5) midpoint: (1,-6)?

Answer 1

Start point `" "->P_s->(x_s,y_s)" "->" "(1,-17)`

Explanation:

The distance from mid to end is the same distance from start to mid
as mid point is the mean point

Let end point be `P_e`

Let mean point be `P_m`

Let start point be `P_s`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Mid to end `-> P_e-P_m`

Start to mid `-> P_m-P_s`

`color(green)("As distance from mid to each end is the same")`

we have `P_m-P_s= P_e-P_m`.......(1)

Multiply equation (1) by (-1) so that `P_s` is positive

`-P_m+P_s = +P_m - P_e`

Add `P_m` to both sides

`P_s=P_m+P_m-P_e`

`P_s=2P_m-P_e`

`P_s->(x_s,y_s)`

`x_s=2(x_m)-x_e" "->" "2(1)-1=1`

`y_s=2(y_m)-y_e" "->" "2(-6)-5=-17`

`P_s->(x_s,y_s)" "->" "(1,-17)`