Question

given the points m(-3, -4) and t(5, 0). find the coordinates of the point q on directed line segment mt that partitions mt in the ratio 2:3.

Explanation:

Step1: Recall the section - formula

The formula to find the coordinates of a point \(Q(x,y)\) that divides the line - segment joining \(M(x_1,y_1)\) and \(T(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-3,y_1=-4,x_2 = 5,y_2 = 0,m = 2,n = 3\).

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

\[

$$\begin{align*} x&=\frac{2\times5+3\times(-3)}{2 + 3}\\ &=\frac{10-9}{5}\\ &=\frac{1}{5}=0.2 \end{align*}$$

\]

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

\[

$$\begin{align*} y&=\frac{2\times0+3\times(-4)}{2 + 3}\\ &=\frac{0 - 12}{5}\\ &=-\frac{12}{5}=-2.4 \end{align*}$$

\]

Answer:

\((0.2,-2.4)\)