Question

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

Explanation:

1. First, assume the coordinates of point \(A=(x_1,y_1)\) and point \(B=(x_2,y_2)\). From the graph, \(A = (-5,-4)\) and \(B=(-3,3)\).

• The formula for finding the coordinates of a point \(P=(x,y)\) that divides the line - segment from \(A=(x_1,y_1)\) to \(B=(x_2,y_2)\) 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, the point is \(\frac{3}{4}\) of the way from \(A\) to \(B\), so \(m = 3\) and \(n = 1\).

1. Calculate the \(x\) - coordinate of the point:

• Substitute \(x_1=-5\), \(x_2=-3\), \(m = 3\), and \(n = 1\) into the \(x\) - coordinate formula \(x=\frac{mx_2+nx_1}{m + n}\).
• \(x=\frac{3\times(-3)+1\times(-5)}{3 + 1}=\frac{-9 - 5}{4}=\frac{-14}{4}=-\frac{7}{2}=-3.5\).

1. Calculate the \(y\) - coordinate of the point:

• Substitute \(y_1=-4\), \(y_2=3\), \(m = 3\), and \(n = 1\) into the \(y\) - coordinate formula \(y=\frac{my_2+ny_1}{m + n}\).
• \(y=\frac{3\times3+1\times(-4)}{3 + 1}=\frac{9-4}{4}=\frac{5}{4}=1.25\).

Answer:

\((-3.5,1.25)\)