How does one derive the Midpoint Formula?

1 Answer
Daniel L.
Nov 23, 2015

It can be prooven using vectors. See explanation.

Explanation:

Let there be 2 points: #A=(x_A;y_A)# and #B=(x_B,y_B)#. We are looking for a point #M# for which vectors #vec(AM)# and #vec(MB)# are equal. Using the equality of vectors we have:

#[x_M-x_A;y_M-y_A]=[x_B-x_M;y_B-y_M]#.

Now we can calculate both coordinates separately:

#x_M-x_A=x_B-x_M#

#x_M+x_M=x_B+x_A#

#2x_M=x_A+x_B#

#x_M=(x_A+x_B)/2#

For #y# coordinate we have similar equation:

#y_M-y_A=y_B-y_M#

#y_M+y_M=y_B+y_A#

#2y_M=y_A+y_B#

#y_M=(y_A+y_B)/2#

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