Question

the coordinates of the endpoints of $overline{ab}$ are $a(3,6)$ and $b(7, - 10)$. point $c$ is on $overline{ab}$ and divides it such that $ac:bc$ is $1:3$. what are the coordinates of $c$? write your answers as integers or decimals.

Explanation:

Step1: Recall the section - formula

The section - formula for 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 given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1 = 3,y_1 = 6,x_2 = 7,y_2=-10,m = 1,n = 3\).

Step2: Calculate the x - coordinate of point C

\[

$$\begin{align*} x&=\frac{1\times7+3\times3}{1 + 3}\\ &=\frac{7 + 9}{4}\\ &=\frac{16}{4}\\ &=4 \end{align*}$$

\]

Step3: Calculate the y - coordinate of point C

\[

$$\begin{align*} y&=\frac{1\times(-10)+3\times6}{1 + 3}\\ &=\frac{-10 + 18}{4}\\ &=\frac{8}{4}\\ &=2 \end{align*}$$

\]

Answer:

\((4,2)\)