Video: Finding the Unknown Coordinates of the Midpoint of a Given Line Segment to Find Their Sum

Given the 𝐴(π‘Ž, 𝑏, 𝑐) is the midpoint of the line segment between 𝐡(9, βˆ’17, 2) and 𝐢(16, βˆ’12, 7), what is π‘Ž + 𝑏 + 𝑐?

02:52

Video Transcript

Given the point capital 𝐴 with coordinates lowercase π‘Ž, lowercase 𝑏, lowercase 𝑐 is the midpoint of the line segment between the point capital 𝐡 nine, negative 17, two and the point capital 𝐢 16, negative 12, seven, what is lowercase π‘Ž 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 𝐡 and capital 𝐢, and we’re given the coordinates of capital 𝐡 and capital 𝐢. 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 π‘₯ one, 𝑦 one, 𝑧 one and π‘₯ two, 𝑦 two, 𝑧 two would be given by π‘₯ one plus π‘₯ 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 𝐡 and capital 𝐢, 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 π‘₯ one, 𝑦 one, and 𝑧 one equal to the coordinates of capital 𝐡 and π‘₯ two, 𝑦 two, 𝑧 two to the coordinates of point capital 𝐢. We can then use this to find the coordinates of capital 𝐴. First, let’s find the π‘₯-coordinate of point capital 𝐴. To do this, we find the average of the π‘₯-coordinates of point capital 𝐡 and capital 𝐢. That’s nine plus 16 all over two. And this is the π‘₯-coordinate of the midpoint of this line segment. It’s going to be the π‘₯-coordinate of point capital 𝐴, which we know is lowercase π‘Ž. 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 𝐡 and capital 𝐢. 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 𝐡 and capital 𝐢. 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 π‘Ž, lowercase 𝑏, and lowercase 𝑐 into our expression. Substituting lowercase π‘Ž is 25 over two, lowercase 𝑏 is negative 29 over two, and lowercase 𝑐 is nine over two, we get lowercase π‘Ž 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.

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