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

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Answer 1 · 2017-02-19T18:24:24

$x_"other end" = 2x_"midpoint" - x_"starting point"$

$y_"other end" = 2y_"midpoint" - y_"starting point"$

Given:

$x_"starting point" = 0$ ,
$y_"starting point" = 0$
$x_"midpoint" = 2$
$y_"midpoint" = -8$

Find: $x_"other end"$ and $y_"other end"$

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

$Deltax = x_"midpoint" - x_"starting point"" [1]"$

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

$x_"other end" = 2Deltax+ x_"starting point"" [2]"$

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

$x_"other end" = 2(x_"midpoint" - x_"starting point")+ x_"starting point"" [3]"$

Use the distributive property:

$x_"other end" = 2x_"midpoint" - 2x_"starting point" + x_"starting point"" [4]"$

Combine like terms:

$x_"other end" = 2x_"midpoint" - x_"starting point"" [5]"$

The same thing is true of the y coordinate:

$y_"other end" = 2y_"midpoint" - y_"starting point"" [6]"$

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

$x_"other end" = 2(2) - 0$

$y_"other end" = 2(-8) - 0$

$x_"other end" = 4$

$y_"other end" = -16$

The other endpoint is $(4,-16)$