Question Video: Finding an Unknown in the Coordinates of a Point Given That It Lies in the π‘₯𝑦-Plane Mathematics

Given that the midpoint of the line segment 𝐴𝐡 lies in the π‘₯𝑦-plane, and the coordinates of 𝐴 and 𝐡 are (βˆ’12, βˆ’9, π‘˜ + 3) and (βˆ’15, βˆ’9, 3π‘˜), respectively, determine the value of π‘˜.

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Video Transcript

Given that the midpoint of the line segment 𝐴𝐡 lies in the π‘₯𝑦-plane and the coordinates of 𝐴 and 𝐡 are negative 12, negative nine, π‘˜ plus three and negative 15, negative nine, three π‘˜, respectively, determine the value of π‘˜.

In this question, we’re given some information about two points, the point 𝐴 and the point 𝐡. We’re told the coordinates of point 𝐴 are negative 12, negative nine, π‘˜ plus three and the coordinates of point 𝐡 are negative 15, negative nine, and three π‘˜. And we can see there’s an unknown value of π‘˜ in these coordinates. We’re also told that the midpoint of the line segment between 𝐴 and 𝐡 lies in the π‘₯𝑦-plane. We need to use all of this information to determine the value of π‘˜.

To answer this question, we’re going to need to recall two pieces of information. First, we’re going to need to recall what it means for a point to lie in the π‘₯𝑦-plane. And we’re also going to need to recall how we find the midpoint of a line segment where we know the endpoints of our line segment. First, we recall that we say the point 𝑃 with coordinates 𝑃 sub one, 𝑃 sub two, and 𝑃 sub three lies in the π‘₯𝑦-plane if 𝑃 sub three is equal to zero. In other words, the π‘₯𝑦-plane consists of all points with 𝑧-coordinate equal to zero. Therefore, in our question, because we’re told the midpoint of the line segment 𝐴𝐡 lies in the π‘₯𝑦-plane, we know that this point must have 𝑧-coordinate equal to zero.

Next, we’re going to need to recall how to find the midpoint between two points. We know to find the midpoint between the two points π‘₯ sub one, 𝑦 sub one, 𝑧 sub one and π‘₯ sub two, 𝑦 sub two, 𝑧 sub two, we need to find the average of each of our coordinates. In other words, the midpoint is given by the point π‘₯ sub one plus π‘₯ sub two all over two, 𝑦 sub one plus 𝑦 sub two all over two, 𝑧 sub one plus 𝑧 sub two all over two. We can combine these with the information given to us in the question to find the value of π‘˜.

First, we’re going to use our formula to find an expression for the midpoint of the line segment 𝐴𝐡. And it’s worth pointing out here the midpoint between two points is exactly the same as the midpoint of a line segment where these two points are the endpoints of this line segment. So to find the midpoint of the line segment 𝐴𝐡, all we need to do is take the average of the coordinates of point 𝐴 and point 𝐡. This gives us the point negative 12 plus negative 15 all over two, negative nine plus negative nine all over two, π‘˜ plus three plus three π‘˜ all over two.

And we can calculate or simplify each of the expressions in our coordinates. We get the point negative 27 over two, negative nine, four π‘˜ plus three all over two. So now we found an expression for the midpoint of our line segment 𝐴𝐡. And remember, in the question, we’re told that this lies in the π‘₯𝑦-plane. And we know that any point that lies in the π‘₯𝑦-plane must have 𝑧-coordinate equal to zero. Therefore, by saying the 𝑧-coordinate equal to zero, we’ll be able to find the value of π‘˜. Doing this, we get the equation four π‘˜ plus three all over two is equal to zero, and we can solve this for π‘˜. We’ll start by multiplying through by two. This gives us four π‘˜ plus three is equal to zero times two, which is, of course, just equal to zero.

Next, we’ll subtract three from both sides of the equation. This gives us that four π‘˜ is equal to negative three. And finally, we’ll divide through by four or multiply through by one-quarter. This gives us that the value of π‘˜ is negative three over four, which is our final answer. Therefore, we were able to show if the midpoint of the line segment 𝐴𝐡 lies in the π‘₯𝑦-plane and the coordinates of 𝐴 are negative 12, negative nine, π‘˜ plus three and the coordinates of 𝐡 are negative 15, negative nine, and three π‘˜, then the value of π‘˜ has to be equal to negative three over four.

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