Question

the coordinates of the endpoints of $overline{gh}$ are $g(2,7)$ and $h(11,16)$. point $i$ is on $overline{gh}$ and divides it such that $gi:hi$ is $4:5$. what are the coordinates of $i$? write your answers as integers or decimals.

Explanation:

Step1: Use the section - formula for x - coordinate

The section formula for the x - coordinate of a point \(I(x,y)\) that divides the line segment joining \(G(x_1,y_1)\) and \(H(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1 = 2\), \(x_2=11\), \(m = 4\), \(n = 5\). So \(x=\frac{4\times11+5\times2}{4 + 5}=\frac{44 + 10}{9}=\frac{54}{9}=6\).

Step2: Use the section - formula for y - coordinate

The section formula for the y - coordinate of a point \(I(x,y)\) that divides the line segment joining \(G(x_1,y_1)\) and \(H(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1 = 7\), \(y_2=16\), \(m = 4\), \(n = 5\). So \(y=\frac{4\times16+5\times7}{4 + 5}=\frac{64+35}{9}=\frac{99}{9}=11\).

Answer:

\((6,11)\)