In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7, 6) ?

Let the point P(5, 4) divide the line segment joining the points (2, 1), (7, 6) in the ratio m₁ : m₂.

By Section-formula, we get the coordinates of point P as:

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

Putting values in x coordinate of above equation we get,

$= \dfrac{m$1 \times 7 + m2 \times 2}{m1 + m2} \\[1em] = \dfrac{7m1 + 2m2}{m1 + m2} \\[1em]

According to question, the x-coordinate of P = 5. Comparing we get,

$\Rightarrow \dfrac{7m$1 + 2m2}{m1 + m2} = 5 \\[1em] \Rightarrow 7m1 + 2m2 = 5m1 + 5m2 \\[1em] \Rightarrow 7m1 - 5m1 = 5m2 - 2m2 \\[1em] \Rightarrow 2m1 = 3m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{3}{2}.

Hence, the ratio in which point (5, 4) divides the line segment is 3 : 2.