Video: Determining the Coordinates of the Midpoint of a Line Segment Given the Coordinates of Its Two Ends

Points 𝐴, 𝐵 have coordinates (8, −8, −12) and (−8, 5, −8) respectively. Determine the coordinates of the midpoint of the line segment 𝐴𝐵.

02:16

Video Transcript

Points 𝐴 and 𝐵 have coordinates eight, negative eight, negative 12 and negative eight, five, negative eight, respectively. Determine the coordinates of the midpoint of the line segment 𝐴𝐵.

We’re looking for the midpoint of a line segment in three dimensions. You might be aware of how to do this in two dimensions. There’s a formula. It turns out that there’s a very similar formula for three dimensions. We’ve phrased this as the midpoint of two points rather than the midpoint of a line segment connecting the two points. But the effect is the same.

The 𝑥-coordinate of the midpoint is the mean of the 𝑥-coordinates of the two points. The same is true for the 𝑦-coordinate. It’s the mean of the 𝑦-coordinates of the two points. And the 𝑧-coordinate, which is new, is the mean of the 𝑧-coordinates of the two points.

The bits in the pinky-purple color might be new to you. But hopefully you’re aware of the midpoint formula in two dimensions, which is what you get if you just get rid of the pinky-purple text. Applying this formula is straightforward.

The 𝑥-coordinate is the mean of the 𝑥-coordinates of points 𝐴 and 𝐵. So it’s eight plus negative eight over two. The 𝑦-coordinate is the mean of the 𝑦-coordinates of points 𝐴 and 𝐵. So it’s negative eight plus five over two. And finally, the 𝑧-coordinate is the mean of the 𝑧-coordinates of points 𝐴 and 𝐵. So that’s negative 12 plus negative eight over two.

And now all we have to do is simplify each coordinate. Eight plus negative eight over two is just zero. Negative eight plus five is negative three. And dividing this by two, we get negative three over two. And finally, negative 12 plus negative eight is negative 20. And dividing this by two, we get negative 10.

These are the coordinates of the midpoint of points 𝐴 and 𝐵. You can also think of this point as being the midpoint of the line segment between 𝐴 and 𝐵, the point that lies on this line segment and divides it neatly into two parts of equal length.

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