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

Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). The formula for finding a point \(C=(x,y)\) that divides the line - segment \(\overline{AB}\) in the ratio \(m:n\) is \(x=x_1+\frac{m}{m + n}(x_2 - x_1)\) and \(y=y_1+\frac{m}{m + n}(y_2 - y_1)\). 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 \(1:4\)).

Step2: Calculate \(x\) - coordinate of \(C\)

\(x=x_1+\frac{1}{1 + 4}(x_2 - x_1)=x_1+\frac{1}{5}(x_2 - x_1)=\frac{5x_1+x_2 - x_1}{5}=\frac{4x_1+x_2}{5}\)

Step3: Calculate \(y\) - coordinate of \(C\)

\(y=y_1+\frac{1}{1 + 4}(y_2 - y_1)=y_1+\frac{1}{5}(y_2 - y_1)=\frac{5y_1+y_2 - y_1}{5}=\frac{4y_1+y_2}{5}\)

Since the coordinates of \(A\) and \(B\) are not given in the problem, the general coordinates of point \(C\) that is \(\frac{1}{5}\) of the distance from \(A=(x_1,y_1)\) to \(B=(x_2,y_2)\) are \((\frac{4x_1+x_2}{5},\frac{4y_1+y_2}{5})\)

Answer:

\(C=(\frac{4x_1+x_2}{5},\frac{4y_1+y_2}{5})\)