Find the coordinates of the points which divide the line

Question:

Find the coordinates of the points which divide the line segment joining A(−2, 2) and B (2, 8) into four equal parts.

Solution:

The co-ordinates of the midpoint $\left(x_{n}, y_{n}\right)$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by,

$\left(x_{m}, y_{m}\right)=\left(\left(\frac{x_{1}+x_{2}}{2}\right),\left(\frac{y_{1}+y_{2}}{2}\right)\right)$

Here we are supposed to find the points which divide the line joining A(−2,2) and B(2,8) into 4 equal parts.

We shall first find the midpoint M(x, y) of these two points since this point will divide the line into two equal parts.

$\left(x_{w}, y_{m}\right)=\left(\left(\frac{-2+2}{2}\right),\left(\frac{2+8}{2}\right)\right)$

So the point M(0,5) splits this line into two equal parts.

Now, we need to find the midpoint of A(−2,2) and M(0,5) separately and the midpoint of B(2,8) and M(0,5). These two points along with M(0,5) split the line joining the original two points into four equal parts.

Let  be the midpoint of A(−2,2) and M(0,5).

$(e, d)=\left(\left(\frac{-2+0}{2}\right),\left(\frac{2+5}{2}\right)\right)$

$(e, d)=\left(-1, \frac{7}{2}\right)$

Now let $M_{2}(g, h)$ bet the midpoint of $B(2,8)$ and $M(0,5)$.

$(g, h)=\left(\left(\frac{2+0}{2}\right),\left(\frac{8+5}{2}\right)\right)$

$(g, h)=\left(1, \frac{13}{2}\right)$

Hence the co-ordinates of the points which divide the line joining the two given points are $\left(-1, \frac{7}{2}\right),(0,5)$ and $\left(1, \frac{13}{2}\right)$.