Find the coordinates of the point that internally divides the line connecting (1, -2) and (4, 7) in the ratio 1: 2.

We know if a point divides the line segment AB joining two points $\mathrm{A}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ and $\mathrm{B}\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ in a given ratio m:n internally, then by section formula, the coordinates of a point are given as

$=\left(\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{ \mathrm{m}+\mathrm{n}},\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{ \mathrm{m}+\mathrm{n}}\right)$

Since the points (1,-2) and (4,7) divide internally in the ratio 1: 2 then the coordinates of the point can be determined as follows:

$(\mathrm{x},\mathrm{y})=\left(\frac{1\times 4+2\times 1}{1+2},\frac{1\times 7+2\times -2}{1+2}\right)$

$(x,y)=\left(\frac{4+2}{3},\frac{7-4}{3}\right)$

$(x,y)=\left(\frac{6}{3},\frac{3}{3}\right)=(2,1)$

Thus, the coordinate of the required point = (2, 1)