Finding the midpoint of a line segment in three dimensions
Defining the formula for the midpoint of a line segment in three-dimensional space
In this lesson we’ll look at how to find the midpoint of a line segment in three dimensions when we’re given the endpoints of the line segment as coordinates in three-dimensional space.
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Midpoint formula
We can use the midpoint formula for three dimensions to find the middle of the line segment that has endpoints ???P_1=(x_1,y_1,z_1)??? and ???P_2=(x_2,y_2,z_2)???, which is
???M=\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2},\frac{{{z}_{1}}+{{z}_{2}}}{2} \right)???
Applying the midpoint formula to three dimensions
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Finding midpoints and endpoints using the midpoint formula for line segments
Example
Find the midpoint of a line segment joining points ???{{P}_{1}}??? and ???{{P}_{2}}???.
???{{P}_{1}}=(4,-6,8)???
???{{P}_{2}}=(4,3,-5)???
We’ll use the midpoint formula for the midpoint ???M??? between points in three dimensions.
???m=\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2},\frac{{{z}_{1}}+{{z}_{2}}}{2} \right)???
where ???{{P}_{1}}=({{x}_{1}},{{y}_{1}},{{z}_{1}})??? and ???{{P}_{2}}=({{x}_{2}},{{y}_{2}},{{z}_{2}})???. We’ll plug in the points we’ve been given, using ???{{P}_{1}}=(4,-6,8)??? and ???{{P}_{2}}=(4,3,-5)???.
???m=\left( \frac{4+4}{2},\frac{-6+3}{2},\frac{8+-5}{2} \right)=\left( \frac{8}{2},\frac{-3}{2},\frac{3}{2} \right)=(4,-1.5,1.5)???
Let’s work through a different type of example.
We can use the midpoint formula for three dimensions to find the middle of the line segment.
Example
Find point ???A??? if ???M??? is the midpoint of ???\overline{AB}???.
???M=(4.5,-3.5,3)???
???B=(2,-4,8)???
Let’s use the midpoint formula and set up what we know.
???M=\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2},\frac{{{z}_{1}}+{{z}_{2}}}{2} \right)???
???(4.5,-3.5,3)=\left( \frac{{{x}_{1}}+2}{2},\frac{{{y}_{1}}+(-4)}{2},\frac{{{z}_{1}}+8}{2} \right)???
Using the fact that these two points are equivalent, we can set up three equations:
???4.5=\frac{{{x}_{1}}+2}{2}???
???-3.5=\frac{{{y}_{1}}+(-4)}{2}???
???3=\frac{{{z}_{1}}+8}{2}???
Solving each equation gives us
???4.5=\frac{{{x}_{1}}+2}{2}???
???2(4.5)={{x}_{1}}+2???
???9={{x}_{1}}+2???
???7={{x}_{1}}???
and
???-3.5=\frac{{{y}_{1}}+(-4)}{2}???
???2(-3.5)={{y}_{1}}-4???
???-7={{y}_{1}}-4???
???-3={{y}_{1}}???
and
???3=\frac{{{z}_{1}}+8}{2}???
???2(3)={{z}_{1}}+8???
???6={{z}_{1}}+8???
???-2={{z}_{1}}???
So the coordinates of point ???A??? are ???(7,-3,-2)???.
