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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.

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.


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)???.


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