Given:
The coordinates of line segment AB are given as follows:
[tex]\begin{gathered} A\text{ \lparen x}_1,y_1)\text{ = \lparen1,3 \rparen} \\ B(x_2,y_2)\text{ = \lparen 7,8 \rparen} \\ m\text{ : n = 2 : 1} \end{gathered}[/tex]
Required:
Coordinate of point which divides the given line segment in a ratio of 2:1.
Assume the required point as P(x,y).
Explanation:
The required coordinate of point P is calculated using the section formula for internal division.
[tex]P(x,y)\text{ = }\frac{mx_2+nx_1}{m+n}\text{ , }\frac{my_2+ny_1}{m+n}[/tex]
Substituting the values in the formula,
[tex]\begin{gathered} P(x,y)\text{ = \lbrack }\frac{2(7)\text{ + 1\lparen1\rparen}}{2+1}\text{ , }\frac{2(8)+1(3)}{2+1}\text{ \rbrack} \\ P(x,y)\text{ = \lbrack }\frac{14\text{ + 1}}{3}\text{ , }\frac{16\text{ + 3}}{3}\text{ \rbrack} \\ P(x,y)\text{ = \lbrack }\frac{15}{3}\text{ , }\frac{19}{3}\text{ \rbrack} \\ P(x,y)\text{ = \lbrack 5 , }\frac{19}{3}\text{ \rbrack} \\ \end{gathered}[/tex]
Answer:
Thus the coordinate of the point P(x,y) which divides the line segment Ab in a ration 2 : 1 is,
[tex]P(x,y)=\text{ \lbrack 5 ,}\frac{19}{3}\operatorname{\rbrack}[/tex]
