Given that point zero, 17, negative
10 is the midpoint of the line segment 𝐴𝐵 and that 𝐴 is the point negative 19,
seven, 14, what are the coordinates of 𝐵?
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 𝑥 one,
𝑦 one, 𝑧 one and 𝑥 two, 𝑦 two, 𝑧 two is 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.
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 𝑥 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 𝑥 one plus 𝑥 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 𝑥
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 𝑥 two, 𝑦 two, and 𝑧 two, the coordinates of our point 𝐵. Let’s start with the
𝑥-coordinate. First, by using our formula, we
know 𝑥 one plus 𝑥 two all over two has to be equal to the 𝑥-coordinate of our
midpoint zero. But remember, 𝑥 one is the
𝑥-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 𝑥 two all over two is equal to zero.
And we can solve this for our value
of 𝑥 sub two. We need to multiply this equation
through by two and then add 19 to both sides. We get 𝑥 sub two is equal to
19. And this is going to be the
𝑥-coordinate of point 𝐵. Let’s now do exactly the same to
find the 𝑦-coordinate of point 𝐵. 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 𝐵. We’ll now do exactly the same to
find the 𝑧-coordinate of point 𝐵. 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
𝑥-, 𝑦-, and 𝑧-coordinates of our point 𝐵.
Therefore, we’ve shown 𝐵 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 𝐵. 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.