Question

what are the coordinates of the point $\frac{3}{4}$ of the way from $a$ to $b$?

Explanation:

Step1: Identify coordinates of A and B

From the graph, \(A=(-5, - 4)\) and \(B=(-3,3)\).

Step2: Use the section - formula for x - coordinate

The formula for the x - coordinate of a point that divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here \(m = 3\), \(n=1\), \(x_1=-5\), \(x_2=-3\). Then \(x=\frac{3\times(-3)+1\times(-5)}{3 + 1}=\frac{-9 - 5}{4}=\frac{-14}{4}=-\frac{7}{2}=-3.5\).

Step3: Use the section - formula for y - coordinate

The formula for the y - coordinate of a point that divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here \(m = 3\), \(n = 1\), \(y_1=-4\), \(y_2=3\). Then \(y=\frac{3\times3+1\times(-4)}{3 + 1}=\frac{9-4}{4}=\frac{5}{4}=1.25\).

Answer:

\((-3.5,1.25)\)