Question

How do you find the coordinates of the other endpoint of a segment with the given endpoint is K(5,1) and the midpoint is M(1,4)?

Answer 1

See a solution process below:

Explanation:

The formula to find the mid-point of a line segment give the two end points is:

`M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)`

Where `M` is the midpoint and the given points are:

`(color(red)(x_1), color(red)(y_1))` and `(color(blue)(x_2), color(blue)(y_2))`

Substituting the values from the points in the problem gives:

`(1, 4) = ((color(red)(5) + color(blue)(x_2))/2 , (color(red)(1) + color(blue)(y_2))/2)`

We can now solve for `color(blue)(x_2)` and `color(blue)(y_2)`

• `color(blue)(x_2)`:

`(color(red)(5) + color(blue)(x_2))/2 = 1`

`color(green)(2) xx (color(red)(5) + color(blue)(x_2))/2 = color(green)(2) xx 1`

`cancel(color(green)(2)) xx (color(red)(5) + color(blue)(x_2))/color(green)(cancel(color(black)(2))) = 2`

`color(red)(5) + color(blue)(x_2) = 2`

`color(red)(5) - color(green)(5) + color(blue)(x_2) = 2 - color(green)(5)`

`0 + color(blue)(x_2) = -3`

`color(blue)(x_2) = -3`

• `color(blue)(y_2)`

`(color(red)(1) + color(blue)(y_2))/2 = 4`

`color(green)(2) xx (color(red)(1) + color(blue)(y_2))/2 = color(green)(2) xx 4`

`cancel(color(green)(2)) xx (color(red)(1) + color(blue)(y_2))/color(green)(cancel(color(black)(2))) = 8`

`color(red)(1) + color(blue)(y_2) = 8`

`color(red)(1) - color(green)(1) + color(blue)(y_2) = 8 - color(green)(1)`

`0 + color(blue)(y_2) = 7`

`color(blue)(y_2) = 7`

The Other End Point Is: `(color(blue)(-3), color(blue)(7))`