Question

the coordinates of the endpoints of $overline{tu}$ are $t(-2,7)$ and $u(4, - 8)$. point $v$ is on $overline{tu}$ and divides it such that $tv:uv$ is $1:2$. what are the coordinates of $v$? 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 \(V(x,y)\) that divides the line - segment joining \(T(x_1,y_1)\) and \(U(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1=-2\), \(x_2 = 4\), \(m = 1\), and \(n = 2\).
\[x=\frac{1\times4+2\times(-2)}{1 + 2}=\frac{4-4}{3}=0\]

Step2: Use the section - formula for y - coordinate

The formula for the y - coordinate of a point \(V(x,y)\) that divides the line - segment joining \(T(x_1,y_1)\) and \(U(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1 = 7\), \(y_2=-8\), \(m = 1\), and \(n = 2\).
\[y=\frac{1\times(-8)+2\times7}{1 + 2}=\frac{-8 + 14}{3}=2\]

Answer:

\((0,2)\)