Find the coordinates of the points which divide the line segment joining the points $(-4, 0)$ and $(0, 6)$ in four equal parts.
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Answer
The co-ordinates of the midpoint $\left(x_m, y_m\right)$ between two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is given by,
$(\text{x}_\text{m},\text{y}_\text{m})=\bigg[\Big(\frac{\text{x}_1+\text{x}_2}{2}\Big),\Big(\frac{\text{y}_1+\text{y}_2}{2}\Big)\bigg]$
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,
$(\text{x}_\text{m},\text{y}_\text{m})=\bigg[\Big(\frac{-4+0}{2}\Big),\Big(\frac{0+6}{2}\Big)\bigg]$
$(\text{x}_\text{m},\text{y}_\text{m})=(-2,3)$
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 $M_1(e, d)$ be the midpoint of $A(-4, 0)$ and $M(-2, 3).$
$(\text{e},\text{d})=\bigg[\Big(\frac{-4-2}{2}\Big),\Big(\frac{0+3}{2}\Big)\bigg]$
$(\text{e},\text{d})=\Big(-3,\frac{3}{2}\Big)$
Now let $M_2(g, h)$ be the midpoint of $B(0, 6)$ and $M(-2, 3).$
$(\text{g},\text{h})=\bigg[\Big(\frac{0-2}{2}\Big),\Big(\frac{6+3}{2}\Big)\bigg]$
$(\text{g},\text{h})=\Big(-1,\frac{9}{2}\Big)$
Hence the co-ordinates of the points which divide the line joining the two given points are $\Big(-3,\frac{3}{2}\Big),$ $(-2, 3)$ and $\Big(-1,\frac{9}{2}\Big).$
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