The endpoints of a line segment AB are given as A(1,7) and B(5,15).
It is required to find the coordinates of a point P along AB so that the ratio of AP to PB is 3 to 1.
To do this, use the Internal Section Formula for a point P dividing the line segment AB with endpoints A(x1,y1) and B(x2,y2) in the ratio m:n.
The formula is given as:
[tex]P(x,y)=\left(\frac{mx_1+nx_2}{m+n},\frac{my_1+ny_2}{m+n}\right)[/tex]Substitute A(x1,y1)=(1,7), B(x2,y2)=(5,15) and m:n=3:1 into the formula:
[tex]\begin{gathered} P(x,y)=\left(\frac{3(1)+1(5)}{3+1},\frac{3(7)+1(15)}{3+1}\right)=\left(\frac{3+5}{3+1},\frac{21+15}{3+1}\right) \\ =\left(\frac{8}{4},\frac{36}{4}\right)=\left(2,9\right) \end{gathered}[/tex]Hence, the required point is P(2,9).
The answer is P(2,9).
