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.
