Question

find the coordinates of point b on $overline{ac}$ such that the ratio of $overline{ab}$ to $overline{bc}$ is 1:4. a(-11, -7) c(12, -5) show your work here

Explanation:

Step1: Recall the section - formula

If a point \(B(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 1\), \(n = 4\), \(x_1=-11\), \(y_1=-7\), \(x_2 = 12\), and \(y_2=-5\).

Step2: Calculate the \(x\) - coordinate of point \(B\)

\[

$$\begin{align*} x&=\frac{1\times12+4\times(-11)}{1 + 4}\\ &=\frac{12-44}{5}\\ &=\frac{-32}{5}=-6.4 \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate of point \(B\)

\[

$$\begin{align*} y&=\frac{1\times(-5)+4\times(-7)}{1 + 4}\\ &=\frac{-5-28}{5}\\ &=\frac{-33}{5}=-6.6 \end{align*}$$

\]

Answer:

\((-6.4,-6.6)\)