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)?

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Answer 1 · 2018-05-23T11:40:14

See a solution process below:

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

$M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ $M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)$

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

$(x_{1}, y_{1})$ $(color(red)(x_1), color(red)(y_1))$ and $(x_{2}, y_{2})$ $(color(blue)(x_2), color(blue)(y_2))$

Substituting the values from the points in the problem gives:

$(1, 4) = (\frac{5 + x_{2}}{2}, \frac{1 + y_{2}}{2})$ $(1, 4) = ((color(red)(5) + color(blue)(x_2))/2 , (color(red)(1) + color(blue)(y_2))/2)$

We can now solve for $x_{2}$ $color(blue)(x_2)$ and $y_{2}$ $color(blue)(y_2)$

$\frac{5 + x_{2}}{2} = 1$ $(color(red)(5) + color(blue)(x_2))/2 = 1$

$2 \times \frac{5 + x_{2}}{2} = 2 \times 1$ $color(green)(2) xx (color(red)(5) + color(blue)(x_2))/2 = color(green)(2) xx 1$

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

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

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

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

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

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

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

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

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

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

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

$color(blue)(y_2) = 7$

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