Question

day 1 - directed line segment
point a (2, 4) and b (12, 14). find the coordinates of p that divides $overline{ab}$ in a ratio of 1:4.

Explanation:

Step1: Recall the section - formula

The formula to find the coordinates of 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 \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1 = 2,y_1 = 4,x_2 = 12,y_2 = 14,m = 1,n = 4\).

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

\[

$$\begin{align*} x&=\frac{1\times12 + 4\times2}{1+4}\\ &=\frac{12 + 8}{5}\\ &=\frac{20}{5}\\ &=4 \end{align*}$$

\]

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

\[

$$\begin{align*} y&=\frac{1\times14+4\times4}{1 + 4}\\ &=\frac{14 + 16}{5}\\ &=\frac{30}{5}\\ &=6 \end{align*}$$

\]

Answer:

The coordinates of \(P\) are \((4,6)\)