How does one derive the Midpoint Formula?

Question

Sadie Adams · Accepted Answer

It can be prooven using vectors. See 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#

Violet Aldrich · Answer

undefined To derive the Midpoint Formula, you can follow these steps:

1. Start with two points in a Cartesian coordinate system, let's call them $(x_1, y_1)$ and $(x_2, y_2)$.

2. To find the midpoint of the line segment connecting these two points, you need to average their respective $x$ coordinates and their respective $y$ coordinates.

3. Average the $x$ coordinates by adding $x_1$ and $x_2$ together and then dividing by 2. This gives you the $x$-coordinate of the midpoint.

4. Similarly, average the $y$ coordinates by adding $y_1$ and $y_2$ together and then dividing by 2. This gives you the $y$-coordinate of the midpoint.

5. Once you have the average of the $x$ and $y$ coordinates, you'll have the coordinates of the midpoint.

So, the Midpoint Formula is:

$$ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points, and the Midpoint is the coordinates of the midpoint of the line segment connecting these two points.