Question

Point P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in five equal parts.
Find the coordinates of the points P, Q and R.

Answer 1

Since, the points P, Q, R and S divide the line segment joining the points A(1, 2) and B(6, 7) in five equal parts, so
AP = PQ = QR = RS = SB
Here, point P divides AB in the ratio of 1 : 4 internally. So using section formula, we get
$$\begin{gathered} \mathrm{Coordinates}\mathrm{of}P=\left(\frac{1\times \left(6\right)+4\times \left(1\right)}{1+4},\frac{1\times \left(7\right)+4\times \left(2\right)}{1+4}\right) \\ =\left(\frac{6+4}{5},\frac{7+8}{5}\right)=\left(2,3\right) \end{gathered}$$
The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get
$$\begin{gathered} \mathrm{Coordinates}\mathrm{of}Q=\left(\frac{2\times \left(6\right)+3\times \left(1\right)}{2+3},\frac{2\times \left(7\right)+3\times \left(2\right)}{2+3}\right) \\ =\left(\frac{12+3}{5},\frac{14+6}{5}\right)=\left(3,4\right) \end{gathered}$$
The point R divides AB in the ratio of 3 : 2 internally. So using section formula, we get
$$\begin{gathered} \mathrm{Coordinates}\mathrm{of}R=\left(\frac{3\times \left(6\right)+2\times \left(1\right)}{3+2},\frac{3\times \left(7\right)+2\times \left(2\right)}{3+2}\right) \\ =\left(\frac{18+2}{5},\frac{21+4}{5}\right)=\left(4,5\right) \end{gathered}$$
Hence, the coordinates of the points P, Q and R are (2, 3), (3, 4) and (4, 5) respectively.