# Find the coordinates of the points which divide the line segment joining

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:

Here, the given points are A(–2, 2) and B(2, 8)

Let $P_{1}, P_{2}$ and $P_{3}$ divide $A B$ in four equal parts.

$\because \quad \mathrm{AP}_{1}=\mathrm{P}_{1} \mathrm{P}_{2}=\mathrm{P}_{2} \mathrm{P}_{3}=\mathrm{P}_{3} \mathrm{~B}$

Obviously, $\mathrm{P}_{2}$ is the mid-point of $\mathrm{AB}$

$\therefore \quad$ Coordinates of $\mathrm{P}_{2}$ are

$\left(\frac{-\boldsymbol{2}+\boldsymbol{2}}{\boldsymbol{2}}, \frac{\boldsymbol{2}+\boldsymbol{8}}{\boldsymbol{2}}\right)$ or $(0,5)$

Again, $\mathrm{P}_{1}$ is the mid-point of $\mathrm{AP}_{2}$.

$\therefore \quad$ Coordinates of $P_{1}$ are

$\left(\frac{-2+0}{2}, \frac{2+5}{2}\right)$ or $\left(-1, \frac{7}{2}\right)$

Also $\mathrm{P}_{3}$ is the mid-point of $\mathrm{P}_{2} \mathrm{~B}$.

$\therefore \quad$ Coordinates of $\mathrm{P}_{3}$ are

$\left(\frac{0+2}{2}, \frac{5+8}{2}\right) \operatorname{or}\left(1, \frac{13}{2}\right)$

Thus, the coordinates of $P_{1}, P_{2}$ and $P_{3}$ are $\left(-\mathbf{1}, \frac{\mathbf{7}}{\mathbf{2}}\right),(0,5)$ and $\left(\mathbf{1}, \frac{\mathbf{3}}{\mathbf{2}}\right)$ respectively.