Question

find the coordinates of the point $\frac{1}{3}$ of the way from $a$ to $b$.
$a(-3,-2)$
$b(12,4)$
coordinates = ( )

Explanation:

Step1: Use the section - formula

The section - formula for a point \(P(x,y)\) 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, \(m = 1\), \(n=2\), \(x_1=-3\), \(y_1=-2\), \(x_2 = 12\), and \(y_2 = 4\).

Step2: Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{1\times12+2\times(-3)}{1 + 2}\\ &=\frac{12-6}{3}\\ &=\frac{6}{3}\\ &=2 \end{align*}$$

\]

Step3: Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{1\times4+2\times(-2)}{1 + 2}\\ &=\frac{4 - 4}{3}\\ &=0 \end{align*}$$

\]

Answer:

\((2,0)\)