Video Transcript
Suppose ๐ด is negative seven,
negative four; ๐ต is six, negative nine; and ๐ท is eight, negative two. If ๐ถ is the midpoint of both line
segment ๐ด๐ต and line segment ๐ท๐ธ, find point ๐ธ.
Letโs first sketch what we
know. We have a line segment ๐ด๐ต with
the midpoint ๐ถ. And ๐ถ is also the midpoint of line
segment ๐ท๐ธ. Weโre given the coordinates for ๐ด,
๐ต, and ๐ท. Our end goal is to find the
coordinates of point ๐ธ. But before we can find ๐ธ, weโll
need to know ๐ถ. Once we find ๐ถ, we can find
๐ธ. And to do both of these things,
weโll need to remember that the midpoint formula looks like this.
The ๐ฅ-coordinate of the midpoint
is found by taking the ๐ฅ-coordinates from the endpoints and dividing by two. And the ๐ฆ-coordinate of the
midpoint is found by averaging the ๐ฆ-coordinates of the two endpoints. Since ๐ถ is the midpoint of ๐ด and
๐ต, weโll let ๐ด be ๐ฅ one, ๐ฆ one and ๐ต be ๐ฅ two, ๐ฆ two. The midpoint ๐ถ will be located at
negative seven plus six over two, negative four plus negative nine over two. Negative seven plus six over two is
negative one-half. And negative four plus negative
nine is negative 13. So, the ๐ฆ-coordinate is negative
13 over two. Now, we know where ๐ถ is
located. And weโre ready to think about
๐ธ.
If ๐ถ is also the midpoint of ๐ท๐ธ,
then the coordinates of ๐ถ will be equal to the ๐ฅ-coordinates of ๐ท and ๐ธ averaged
together and the ๐ฆ-coordinates of ๐ท and ๐ธ averaged together. Weโre given the coordinates of
๐ท. Thatโs eight, negative two. And so, we plug that in. From here, weโll make two separate
equations. Weโll set negative one-half equal
to eight plus the ๐ฅ-coordinate of ๐ธ over two. And negative 13 over two is equal
to negative two plus the ๐ฆ-coordinate of ๐ธ over two. Weโll give ourselves a little bit
more room.
Since all of the denominators are
two, then the numerators are equal to each other. Negative one equals eight plus the
๐ฅ-coordinate of ๐ธ. And to solve for that missing
value, we subtract eight from both sides. And we see the ๐ฅ-coordinate for
point ๐ธ is negative nine. To solve for the ๐ฆ-coordinate of
point ๐ธ, we add two to both sides. The ๐ฆ-coordinate of ๐ธ is negative
11. In coordinate form, point ๐ธ is
located at negative nine, negative 11.
