Given a segment joining the points A = (1, 10) and B = (20, 18).
To find the point (a, b) that partitions the segment AB into a 2:5 ratio, we use the equations:
[tex]\begin{gathered} \frac{a-1}{20-a}=\frac{2}{5}\ldots(1) \\ \frac{b-10}{18-b}=\frac{2}{5}\ldots(2) \end{gathered}[/tex]That is, the 2:5 ratio also holds for the x and y coordinates. Solving equation (1) for a:
[tex]\begin{gathered} 5(a-1)=2(20-a) \\ 5a-5=40-2a \\ 7a=45 \\ a=\frac{45}{7}=6\frac{3}{7} \end{gathered}[/tex]Now, solving equation (2) for b:
[tex]\begin{gathered} 5(b-10)=2(18-b) \\ 5b-50=36-2b \\ 7b=86 \\ b=\frac{86}{7}=12\frac{2}{7} \end{gathered}[/tex]So the point is:
[tex](6\frac{3}{7},12\frac{2}{7})[/tex]