M is the mid-point of the line segment joining the points A(0, 4) and B(6, 0). M also divides the line segment OP in the ratio 1 : 3. Find :

(i) co-ordinates of M

(ii) co-ordinates of P

(iii) length of BP

(i) By formula,

Mid-point (M) = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get,

$M = \Big(\dfrac{0 + 6}{2}, \dfrac{4 + 0}{2}\Big) \\[1em] = \Big(\dfrac{6}{2}, \dfrac{4}{2}\Big) \\[1em] = (3, 2).$

Hence, M = (3, 2).

(ii) Let co-ordinates of P be (x, y).

Given, M divides the line segment OP in the ratio 1 : 3.

By section formula,

$x = \dfrac{m$1x2 + m2x1}{m1 + m2}

Substituting values we get,

$\Rightarrow 3 = \dfrac{1 \times x + 3 \times 0}{1 + 3} \\[1em] \Rightarrow 3 = \dfrac{x}{4} \\[1em] \Rightarrow x = 12. \\[1em] y = \dfrac{m$1y2 + m2y1}{m1 + m2}

Substituting values we get,

$\Rightarrow 2 = \dfrac{1 \times y + 3 \times 0}{1 + 3} \\[1em] \Rightarrow 2 = \dfrac{y}{4} \\[1em] \Rightarrow y = 8.$

P = (x, y) = (12, 8).

Hence, co-ordinates of P = (12, 8).

(iii) Distance between two points = $\sqrt{(x$2 - x1)^2 + (y2 - y1)^2}

$BP = \sqrt{(12 - 6)^2 + (8 - 0)^2} \\[1em] = \sqrt{(6)^2 + (8)^2} \\[1em] = \sqrt{36 + 64} \\[1em] = \sqrt{100} = 10 \text{ units}. \\[1em]$

Hence, BP = 10 units.