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Question Video: Finding the Coordinates of a Point on a Line Segment given the Coordinates of the Midpoint and the Coordinates of a Second Point Mathematics

Given that point (0, 17, βˆ’10) is the midpoint of the line segment 𝐴𝐡 and that 𝐴 (βˆ’19, 7, 14), what are the coordinates of 𝐡?

03:37

Video Transcript

Given that point zero, 17, negative 10 is the midpoint of the line segment 𝐴𝐡 and that 𝐴 is the point negative 19, seven, 14, what are the coordinates of 𝐡?

In this question, we’re given the coordinates of the midpoint of a line segment. And we’re also told one of the endpoints of this line segment. We need to use this information to determine the coordinates of the other endpoint of this line segment. To do this, we need to recall the formula for finding the midpoint of a line segment. Remember, the midpoint of a line segment is the point equidistant from the endpoints of our line segment. And it’s the same as the midpoint between the two endpoints of our line segment. And we know how to find this. The midpoint of the points π‘₯ one, 𝑦 one, 𝑧 one and π‘₯ two, 𝑦 two, 𝑧 two is 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.

All we do is take the average of each of the coordinates. In this question, we’re given one of the endpoints and we’re given the midpoint. We need to use this to find the other endpoint. To do this, let’s first fill in the values we know. First, we know the coordinates of point 𝐴 are negative 19, seven, 14. So we can set π‘₯ one equal to negative 19, 𝑦 one equal to seven, and 𝑧 one equal to 14. Similarly, the question tells us the coordinates of our midpoint. It has coordinates zero, 17, negative 10. So we know that π‘₯ one plus π‘₯ two all over two is equal to zero, 𝑦 one plus 𝑦 two all over two is equal to 17, and 𝑧 one plus 𝑧 two all over two is equal to negative 10.

But we also know the values of π‘₯ one, 𝑦 one, and 𝑧 one in this case. So we can just substitute these into our formula. This will then give us an equation for π‘₯ two, 𝑦 two, and 𝑧 two, the coordinates of our point 𝐡. Let’s start with the π‘₯-coordinate. First, by using our formula, we know π‘₯ one plus π‘₯ two all over two has to be equal to the π‘₯-coordinate of our midpoint zero. But remember, π‘₯ one is the π‘₯-coordinate of our point 𝐴. This is going to be equal to negative 19. So we can substitute this in, and we get negative 19 plus π‘₯ two all over two is equal to zero.

And we can solve this for our value of π‘₯ sub two. We need to multiply this equation through by two and then add 19 to both sides. We get π‘₯ sub two is equal to 19. And this is going to be the π‘₯-coordinate of point 𝐡. Let’s now do exactly the same to find the 𝑦-coordinate of point 𝐡. First, we’ll set 𝑦 one plus 𝑦 two all over two equal to the 𝑦-coordinate of our midpoint 17. This gives us the following equation. Next, remember that 𝑦 one is the 𝑦-coordinate of point 𝐴, and we know this is equal to seven. So we substitute this in giving us seven plus 𝑦 two all over two is equal to 17.

Then we just need to solve this equation for 𝑦 sub two. We multiply both sides of our equation through by two and then subtract seven from both sides. This gives us that 𝑦 two is equal to 17 times two minus seven which we can calculate is equal to 27. So we’ve now also found the 𝑦-coordinate of point 𝐡. We’ll now do exactly the same to find the 𝑧-coordinate of point 𝐡. This time, we get that 𝑧 one plus 𝑧 two all over two should be equal to the 𝑧-coordinate of our midpoint. That’s negative 10. This gives us the following equation. And we also know from our formula 𝑧 one will be the 𝑧-coordinate of our point 𝐴, which we know is 14.

So we substitute this in. We get 14 plus 𝑧 two all over two should be equal to negative 10. And then we need to solve this equation for 𝑧 two. We’re going to multiply through by two and then subtract 14 from both sides of the equation. We get 𝑧 two is negative 10 times two minus 14, which we can calculate is negative 34, and these three values, the π‘₯-, 𝑦-, and 𝑧-coordinates of our point 𝐡.

Therefore, we’ve shown 𝐡 must have coordinates 19, 27, negative 34. And it’s worth pointing out here we can check our answer by just finding the midpoint between point 𝐴 and point 𝐡. All we would want to do is check the average of the coordinates of these two points and check that we get the point zero, 17, negative 10.

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