A(2, 5), B(–1, 2) and C(5, 8) are the vertices of a ΔABC, M is a point on AB such that AM : MB = 1 : 2. Find the co-ordinates of M. Hence, find the equation of the line passing through the points C and M.

Given AM : MB = 1 : 2. By section-formula the coordinates of M are,

$\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2} \Big)

Substituting values we get,

$M = \Big(\dfrac{1(-1) + 2(2)}{1 + 2}, \dfrac{1(2) + 2(5)}{1 + 2} \Big) \\[1em] = \Big(\dfrac{-1 + 4}{3}, \dfrac{2 + 10}{3} \Big) \\[1em] = \Big(\dfrac{3}{3}, \dfrac{12}{3} \Big) \\[1em] = (1, 4).$

Equation of line CM can be given by two-point formula i.e.,

$y - y$1 = \dfrac{y2 - y1}{x2 - x1}(x - x1)

Substituting values we get,

⇒ y − 8 = $\dfrac{4 - 8}{1 - 5}$(x − 5)

⇒ y − 8 = $\dfrac{-4}{-4}$(x − 5)

⇒ y − 8 = 1(x − 5)

⇒ y − 8 = x − 5

⇒ x − y − 5 + 8 = 0

⇒ x − y + 3 = 0.

Hence, the equation of CM is x - y + 3 = 0 and the coordinates of M are (1, 4).