Question

Use the Midpoint Formula three times to find the three points that divide the line segment joining (x1, y1) and (x2, y2) into four parts

Answer 1

We are given points $A(x_1,y_1)$ and $B(x_2,y_2)$. 

We first find the midpoint M, of AB, which divides the segment AB into 2 equal parts, 

then we find the midpoint N of AM, and midpoint K of MB.

Thus each of the half parts is divided into 2 equal parts. The whole segment is divided into 4 equal parts.




The coordinates of M, N and K are found as follows:


the coordinates of M are: $( \frac{x_1+x_2}{2} , \frac{y_1+y_2}{2})$


the coordinates of N are: 

$\displaystyle{( \frac{x_1+\frac{x_1+x_2}{2}}{2} , \frac{y_1+\frac{y_1+y_2}{2}}{2})=( \frac{\frac{2x_1+x_1+x_2}{2}}{2} , \frac{\frac{2y_1+y_1+y_2}{2}}{2})$

$=\displaystyle{(\frac{3x_1+x_2}{4}, \frac{3y_1+y_2}{4})}$


similarly, the coordinates of k are:

$=\displaystyle{(\frac{x_1+3x_2}{4}, \frac{y_1+3y_2}{4})}$