Question

Find the coordinates of the points which divide the line segment joining the points (-4, 0) and (0, 6) in four equal parts.

Answer 1

The co-ordinates of the midpoint ${\displaystyle (x_m,y_m)}$ between two points ${\displaystyle (x_1,y_1)}$ and (x_2, y_2) is given by,

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

Here we are supposed to find the points which divide the line joining A(-4,0) and B(0,6) 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

${\displaystyle (x_m,y_m)=\left(\left(\frac{-4+0}{2}\right)\text{,}\left(\frac{0+6}{2}\right)\right)}$

${\displaystyle (x_m,y_m)=(-2,3)}$

So the point M(-2,3) splits this line into two equal parts.

Now, we need to find the midpoint of A(-4,0) and M(-2,3) separately and the midpoint of B(0,6) and M(-2,3). These two points along with M(-2,3) split the line joining the original two points into four equal parts.

Let M(e, d) be the midpoint of A(-4,0) and M(-2,3).

${\displaystyle (e,d)=\left(\left(\frac{-4-2}{2}\right)\text{,}\left(\frac{0+3}{2}\right)\right)}$

${\displaystyle (e,d)=(-3,\frac{3}{2})}$

Now let ${\displaystyle M_2(g,h)}$ bet the midpoint of B(0,6) and M(-2,3).

${\displaystyle (g,h)=\left(\frac{0-2}{2}\right)\text{,}\left(\frac{6+3}{2}\right)}$

${\displaystyle (g,h)=(-1,\frac{9}{2})}$

Hence the co-ordinates of the points which divide the line joining the two given points are (-3,3/2), (-2, 3) and (-1, 9/2).