Points A, B, C and D divide the line segment joining the points (5, -10) and the origin in five equal parts. Find the co-ordinates of B and D.

Let point P = (5, -10) and origin (O) = (0, 0).

Points A, B, C and D divide the line segment PO in 5 equal parts.

From figure,

B divides the line segment PO in the ratio 2 : 3.

Let B be (a, b).

By formula,

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

Substituting values we get,

$\Rightarrow a = \dfrac{2 \times 0 + 3 \times 5}{2 + 3} \\[1em] \Rightarrow a = \dfrac{0 + 15}{5} \\[1em] \Rightarrow a = \dfrac{15}{5} = 3.$

$y = \dfrac{m$1y2 + m2y1}{m1 + m2}

Substituting values we get,

$\Rightarrow b = \dfrac{2 \times 0 + 3 \times -10}{2 + 3} \\[1em] \Rightarrow b = \dfrac{0 - 30}{5} \\[1em] \Rightarrow b = -\dfrac{30}{5} = -6.$

B = (a, b) = (3, -6).

D divides the line segment PO in the ratio 4 : 1.

Let D be (c, d).

By formula,

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

Substituting values we get,

$\Rightarrow c = \dfrac{4 \times 0 + 1 \times 5}{4 + 1} \\[1em] \Rightarrow c = \dfrac{0 + 5}{5} \\[1em] \Rightarrow c = \dfrac{5}{5} = 1.$

$y = \dfrac{m$1y2 + m2y1}{m1 + m2}

Substituting values we get,

$\Rightarrow d = \dfrac{4 \times 0 + 1 \times -10}{4 + 1} \\[1em] \Rightarrow d = -\dfrac{10}{5} \\[1em] \Rightarrow d = -2.$

D = (c, d) = (1, -2).

Hence, B = (3, -6) and D = (1, -2).