Question

the coordinates of the endpoints of $overline{ab}$ are $a(-12, 10)$ and $b(-5, -11)$. point $c$ is on $overline{ab}$ and divides it such that $ac:bc$ is $2:5$. what are the coordinates of $c$? write your answers as integers or decimals.

Explanation:

Step1: Use section - formula for x - coordinate

The section formula for the x - coordinate of a point \(C(x,y)\) that divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1=-12\), \(x_2=-5\), \(m = 2\), \(n = 5\).
\[x=\frac{2\times(-5)+5\times(-12)}{2 + 5}=\frac{-10-60}{7}=\frac{-70}{7}=-10\]

Step2: Use section - formula for y - coordinate

The section formula for the y - coordinate of a point \(C(x,y)\) that divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1 = 10\), \(y_2=-11\), \(m = 2\), \(n = 5\).
\[y=\frac{2\times(-11)+5\times10}{2 + 5}=\frac{-22 + 50}{7}=\frac{28}{7}=4\]

Answer:

\((-10,4)\)