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

May 20, 2016

Start point $\text{ "->P_s->(x_s,y_s)" "->" } \left(1 , - 17\right)$

#### 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 $\to {P}_{e} - {P}_{m}$

Start to mid $\to {P}_{m} - {P}_{s}$

$\textcolor{g r e e n}{\text{As distance from mid to each end is the same}}$

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

${P}_{s} = {P}_{m} + {P}_{m} - {P}_{e}$

${P}_{s} = 2 {P}_{m} - {P}_{e}$

${P}_{s} \to \left({x}_{s} , {y}_{s}\right)$

${x}_{s} = 2 \left({x}_{m}\right) - {x}_{e} \text{ "->" } 2 \left(1\right) - 1 = 1$

${y}_{s} = 2 \left({y}_{m}\right) - {y}_{e} \text{ "->" } 2 \left(- 6\right) - 5 = - 17$

${P}_{s} \to \left({x}_{s} , {y}_{s}\right) \text{ "->" } \left(1 , - 17\right)$