Derive Midpoint Formula

Derive the midpoint formula.

 

The Distance Formula itself is actually derived from the Pythagorean Theorem which is
a^2 + b^2= c^2
where c is the longest side of a right triangle (also known as the hypotenuse) and aa and bb are the other shorter sides (known as the legs of a right triangle).
The very essence of the Distance Formula is to calculate the length of the hypotenuse of the right triangle which is represented by the letter c.

Derivation of the Distance Formula
Suppose you’re given two arbitrary points A and B in the Cartesian plane and you want to find the distance between them.


First, construct the vertical and horizontal line segments passing through each of the given points such that they meet at the 90-degree angle.





Next, connect points A and B to reveal a right triangle.




Find the legs of the right triangle by subtracting the x-values and the y-values accordingly.




The distance between the points A and B is just the hypotenuse of the right triangle.

Note: Hypotenuse is always the side opposite the 90-degree angle.




Finally, applying the concept of the Pythagorean Theorem, the Distance Formula is calculated as follows:

 

Like always, a job well-done!

 

You don't need the Pythagorean theorem to prove this.
Instead 54drop a perpendicular from the midpoint to the horizontal line from x= x1 to x= x2. To the left of that line is a second right triangle with the same angles so we have "similar triangles". The hypotenuse of one is half the length of the other so the two legs of one are half the length of the legs of the other.

 

Thank you everyone.