Consider the points 𝐴: 𝑥, seven;
𝐵: negative four, 𝑦; and 𝐶: two, five. Given that 𝐶 is the midpoint of
line segment 𝐴𝐵, find the values of 𝑥 and 𝑦.
First, let’s list out what we
know. We have line segment 𝐴𝐵 and 𝐶 is
the midpoint point. Point 𝐴 is located at 𝑥,
seven. Point 𝐵 is located at negative
four, 𝑦. And point 𝐶 is located at two,
five. At this point, you might be
thinking, “Shouldn’t we try to graph these values?” But because we’re missing this 𝑥-
and 𝑦-value, it’s not easy to graph this. So, let’s consider what we know
about the midpoint.
The coordinates of the midpoint can
be found by averaging the 𝑥-coordinates of the endpoints and the 𝑦-coordinates of
the endpoints. And if the midpoint is two, five,
then 𝑥 one plus 𝑥 two divided by two has to be equal to two. And 𝑦 one plus 𝑦 two divided by
two has to be equal to five. So, we set up two separate
equations, one that says 𝑥 one plus 𝑥 two over two equals two and one that says
five equals 𝑦 one plus 𝑦 two over two. We’ll let 𝐴 be 𝑥 one, 𝑦 one and
𝐵 be 𝑥 two, 𝑦 two and then we plug in what we know.
Two is then equal to 𝑥 plus
negative four divided by two and five is equal to seven plus 𝑦 divided by two. And now, we just need to solve for
each variable. On the left, we multiply both sides
of the equation by two, which will give us four equals 𝑥 plus negative four, which
we can rewrite to say four equals 𝑥 minus four. Then add four to both sides. And we see that eight equals 𝑥 or,
more commonly, 𝑥 equals eight. We follow the same procedure to
solve for 𝑦. Multiply by two. Subtract seven from both sides. Three equals 𝑦. So, 𝑦 equals three. Since 𝐴 equals 𝑥 seven and 𝑥
equals eight, point 𝐴 is located at eight, seven. And since 𝐵 was located at
negative four, 𝑦 and 𝑦 equals three, 𝐵 is located at negative four, three.