Question

the coordinates of the endpoints of $overline{gh}$ are $g(2, 16)$ and $h(17, 7)$. point $i$ is on $overline{gh}$ and divides it such that $gi:hi$ is $2:1$. what are the coordinates of $i$? write your answers as integers or decimals.

Explanation:

Step1: Use the section - formula for x - coordinate

The 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, \(x_1 = 2\), \(x_2=17\), \(m = 2\), \(n = 1\).
\[x=\frac{2\times17+1\times2}{2 + 1}=\frac{34 + 2}{3}=\frac{36}{3}=12\]

Step2: Use the section - formula for y - coordinate

The 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, \(y_1 = 16\), \(y_2 = 7\), \(m = 2\), \(n = 1\).
\[y=\frac{2\times7+1\times16}{2 + 1}=\frac{14 + 16}{3}=\frac{30}{3}=10\]

Answer:

\((12,10)\)