Question

refer to the coordinate grid.
⑦ find point c on $overline{ab}$ that is $\frac{1}{5}$ of the distance from a to b.

Explanation:

Step1: Assume coordinates of A and B

Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). From the grid, assume \(A = (- 3,4)\) and \(B=(2, - 1)\).

Step2: Use the section - formula

The formula for a point \(C=(x,y)\) 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, \(m = 1\) and \(n = 4\) (since \(C\) is \(\frac{1}{5}\) of the distance from \(A\) to \(B\), so the ratio of \(AC\) to \(CB\) is \(1:4\)).
For the \(x\) - coordinate of \(C\):
\[x=\frac{1\times2+4\times(-3)}{1 + 4}=\frac{2-12}{5}=\frac{-10}{5}=-2\]
For the \(y\) - coordinate of \(C\):
\[y=\frac{1\times(-1)+4\times4}{1 + 4}=\frac{-1 + 16}{5}=\frac{15}{5}=3\]

Answer:

\(C=(-2,3)\)