Question

Points P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R. [CBSE 2014]

Answer 1

It is given that P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts.

∴ AP = PQ = QR = RS = SB .....(1)

Now,

AP + PQ + QR + RS + SB = AB

⇒ AP + AP + AP + AP + AP = AB [From (1)]

⇒ 5AP = AB

⇒ AP = $\frac{1}{5}$AB .....(2)

Now,

PB = PQ + QR + RS + SB = $\frac{1}{5}$AB +$\frac{1}{5}$AB + $\frac{1}{5}$AB + $\frac{1}{5}$AB = $\frac{4}{5}$AB .....(3)

From (2) and (3), we get

AP : PB = $\frac{1}{5}$AB : $\frac{4}{5}$AB = 1 : 4

Similarly,

AQ : QB = 2 : 3 and AR : RB = 3 : 2

Using section formula, we get

Coordinates of P = $\left(\frac{1\times 6+4\times 1}{1+4},\frac{1\times 7+4\times 2}{1+4}\right)=\left(\frac{10}{5},\frac{15}{5}\right)=\left(2,3\right)$

Coordinates of Q = $\left(\frac{2\times 6+3\times 1}{2+3},\frac{2\times 7+3\times 2}{2+3}\right)=\left(\frac{15}{5},\frac{20}{5}\right)=\left(3,4\right)$

Coordinates of R = $\left(\frac{3\times 6+2\times 1}{3+2},\frac{3\times 7+2\times 2}{3+2}\right)=\left(\frac{20}{5},\frac{25}{5}\right)=\left(4,5\right)$