# How do you find the coordinates of the other endpoint of a segment with the given Endpoint: (0, 0); Midpoint: ( 2, -8)?

Feb 19, 2017

${x}_{\text{other end" = 2x_"midpoint" - x_"starting point}}$

${y}_{\text{other end" = 2y_"midpoint" - y_"starting point}}$

#### Explanation:

Given:

${x}_{\text{starting point}} = 0$,
${y}_{\text{starting point}} = 0$
${x}_{\text{midpoint}} = 2$
${y}_{\text{midpoint}} = - 8$

Find: ${x}_{\text{other end}}$ and ${y}_{\text{other end}}$

The change in x, $\Delta x$, from the starting point to the midpoint is:

$\Delta x = {x}_{\text{midpoint" - x_"starting point"" [1]}}$

The other end must be twice that change relative to the starting point:

${x}_{\text{other end" = 2Deltax+ x_"starting point"" [2]}}$

Substitute the right side of equation [1] into equation [2]:

${x}_{\text{other end" = 2(x_"midpoint" - x_"starting point")+ x_"starting point"" [3]}}$

Use the distributive property:

${x}_{\text{other end" = 2x_"midpoint" - 2x_"starting point" + x_"starting point"" [4]}}$

Combine like terms:

${x}_{\text{other end" = 2x_"midpoint" - x_"starting point"" [5]}}$

The same thing is true of the y coordinate:

${y}_{\text{other end" = 2y_"midpoint" - y_"starting point"" [6]}}$

Substituting the given information into equations [5] and [6]:

${x}_{\text{other end}} = 2 \left(2\right) - 0$

${y}_{\text{other end}} = 2 \left(- 8\right) - 0$

${x}_{\text{other end}} = 4$

${y}_{\text{other end}} = - 16$

The other endpoint is $\left(4 , - 16\right)$