Video Transcript
Given the point capital 𝐴 with
coordinates lowercase 𝑎, lowercase 𝑏, lowercase 𝑐 is the midpoint of the line
segment between the point capital 𝐵 nine, negative 17, two and the point capital 𝐶
16, negative 12, seven, what is lowercase 𝑎 plus lowercase 𝑏 plus lowercase
𝑐?
In this question, we’re asked to
find the sum of the coordinates of the point capital 𝐴. And to do this, we’re told that
capital 𝐴 is the midpoint of the line segment between the points capital 𝐵 and
capital 𝐶, and we’re given the coordinates of capital 𝐵 and capital 𝐶. So all we need to do to find the
coordinates of capital 𝐴 is find the midpoint of this line segment. And we know how to find the
midpoint of a line segment. All we need to do is take the
average value of the coordinates of our endpoints.
In other words, the midpoint of the
points 𝑥 one, 𝑦 one, 𝑧 one and 𝑥 two, 𝑦 two, 𝑧 two would be given by 𝑥 one
plus 𝑥 two all over two, 𝑦 one plus 𝑦 two all over two, 𝑧 one plus 𝑧 two all
over two. And since capital 𝐴 is the
midpoint of the line segment between points capital 𝐵 and capital 𝐶, we can use
this formula to find equations for the coordinates of capital 𝐴.
So the first thing we’re going to
want to do is set the values of 𝑥 one, 𝑦 one, and 𝑧 one equal to the coordinates
of capital 𝐵 and 𝑥 two, 𝑦 two, 𝑧 two to the coordinates of point capital 𝐶. We can then use this to find the
coordinates of capital 𝐴. First, let’s find the 𝑥-coordinate
of point capital 𝐴. To do this, we find the average of
the 𝑥-coordinates of point capital 𝐵 and capital 𝐶. That’s nine plus 16 all over
two. And this is the 𝑥-coordinate of
the midpoint of this line segment. It’s going to be the 𝑥-coordinate
of point capital 𝐴, which we know is lowercase 𝑎. And we can calculate this. It’s equal to 25 over two.
Let’s now do exactly the same for
the 𝑦-coordinate of our midpoint. We take the average of the
𝑦-coordinates of point capital 𝐵 and capital 𝐶. This gives us negative 17 plus
negative 12 all over two. And once again, this is going to
give us the 𝑦-coordinate of our midpoint, which is lowercase 𝑏. And we can just calculate this. We get lowercase 𝑏 is negative 29
divided by two. We need to do this one more time,
this time for the 𝑧-coordinate. First, we take the average of the
𝑧-coordinates of capital 𝐵 and capital 𝐶. That’s two plus seven all over
two. And we need to set this equal to
the 𝑧-coordinate of our midpoint, which we’re told is lowercase 𝑐. And once again, we calculate this
expression. This time, we get lowercase 𝑐 is
nine over two.
Now, we can substitute our values
for lowercase 𝑎, lowercase 𝑏, and lowercase 𝑐 into our expression. Substituting lowercase 𝑎 is 25
over two, lowercase 𝑏 is negative 29 over two, and lowercase 𝑐 is nine over two,
we get lowercase 𝑎 plus lowercase 𝑏 plus lowercase 𝑐 is equal to 25 over two plus
negative 29 over two plus nine over two. And if we calculate this
expression, we see that it’s equal to five over two, which is our final answer.
Therefore, given the two endpoints
of a line segment, we were able to find the sum of the coordinates of this
midpoint. All we had to do was take the
average of each of the coordinates of our endpoints and then add these together.
