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

Question:

Point PQR 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 PQ and R.

Solution:

Since, the points PQR 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

Coordinates of $P=\left(\frac{1 \times(6)+4 \times(1)}{1+4}, \frac{1 \times(7)+4 \times(2)}{1+4}\right)$

$=\left(\frac{6+4}{5}, \frac{7+8}{5}\right)=(2,3)$

The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get

Coordinates of $Q=\left(\frac{2 \times(6)+3 \times(1)}{2+3}, \frac{2 \times(7)+3 \times(2)}{2+3}\right)$

$=\left(\frac{12+3}{5}, \frac{14+6}{5}\right)=(3,4)$

The point R divides AB in the ratio of 3 : 2 internally. So using section formula, we get

Coordinates of $R=\left(\frac{3 \times(6)+2 \times(1)}{3+2}, \frac{3 \times(7)+2 \times(2)}{3+2}\right)$

$=\left(\frac{18+2}{5}, \frac{21+4}{5}\right)=(4,5)$

Hence, the coordinates of the points PQ and are (2, 3), (3, 4) and (4, 5) respectively.