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Co-ordinate Geometry — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsCo-ordinate Geometry5 Marks
Question
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,
$\left(x_{m}, y_{m}\right)=\left[\left(\frac{x_1+x_2}{2}\right),\left(\frac{y_1+y_2}{2}\right)\right]$
Here we are supposed to find the points which divide the line joining $A(-4,0)$ 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, $\left( x _{ m }, y _{ m }\right)=\left[\left(\frac{-4+0}{2}\right),\left(\frac{0+6}{2}\right)\right]\left( x _{ m }, y _{ m }\right)=(-2,3)$ 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 $M(-2,3) .(e, d)=\left[\left(\frac{-4-2}{2}\right),\left(\frac{0+3}{2}\right)\right](e, d)=\left(-3, \frac{3}{2}\right)$ Now let $M_2(g, h )$ be the midpoint of $B(0,6)$ and $M(-2,3) \cdot(g, h)=\left[\left(\frac{0-2}{2}\right),\left(\frac{6+3}{2}\right)\right]( g , h )=\left(-1, \frac{9}{2}\right)$ Hence the co-ordinates of the points which divide the line joining the two given points are $\left(-3, \frac{3}{2}\right),(-2,3)$ and $\left(-1, \frac{9}{2}\right)$.

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