Question

given $overline{ac}$ with $a(3, 4)$ and $c(-9, -2)$, if $b$ partitions $ac$ such that the ratio of $ab$ to $bc$ is $1:5$, find the coordinates of $b$.

Explanation:

Step1: Recall section - formula

The formula to find the coordinates of a point \(B(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1 = 3,y_1 = 4,x_2=-9,y_2=-2,m = 1,n = 5\).

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

\[

$$\begin{align*} x&=\frac{1\times(-9)+5\times3}{1 + 5}\\ &=\frac{-9 + 15}{6}\\ &=\frac{6}{6}\\ &=1 \end{align*}$$

\]

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

\[

$$\begin{align*} y&=\frac{1\times(-2)+5\times4}{1 + 5}\\ &=\frac{-2+20}{6}\\ &=\frac{18}{6}\\ &=3 \end{align*}$$

\]

Answer:

The coordinates of \(B\) are \((1,3)\)