Cartesian Coordinate System  Distance and Intermediate Position Between Two Points
Distance and intermediate position between two point in a cartesian x and y coordinate system.
In the rectangular Cartesian coordinate system the coordinate axes are perpendicular to one another and the same unit length is chosen on the two axes.
The distance between two points in Cartesian x and y coordinate system can be calculated as
$$ d(P_1, P_2) = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} \tag{1}$$
Example  Distance between two points
For point P_{1} the coordinates are x_{1} = 4, y_{1} = 5, and for point P_{2} the coordinates are x_{2} = 7, y_{2} = 9.
The distance between the points can be calculated as
$$ d(P_1, P_2) = \sqrt{(7  4)^2 + (9  5)^2} = \underline {5} $$
Intermediate position between two points
The intermediate position between two point in the cartesian coordinate system can be calculated as
$$ x ={ (r_1 x_1 + r_2 x_2) \over (r_1 + r_2 )} \tag{2}$$
$$ y = {(r_1 y_1 + r_2 y_2) \over (r_1 + r_2 ) }\tag{3}$$
where
r_{1} = ratio of the distance between P_{1} to P  to the distance of P_{1} to P_{2}
r_{2} = ratio of the distance between P_{2} to P  to the distance of P_{1} to P_{2}
For the midpoint between P_{1} and P_{2}:
$$ r_1 = r_2 = 1 \tag{4}$$
 and eq. 2 and 3 can be expressed as
$$ x ={ (x_1 + x_2) \over 2} \tag{2a}$$
$$ y = {(y_1 + y_2) \over 2}\tag{3a}$$
Example  Midpoint on Line
The midpoint on the line in the example above can be calculated as
$$ x ={ (4 + 7) \over 2 } = \underline {5.5} $$
$$ y = {(5 + 9) \over 2 } = \underline {7} $$
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