Question

refer to the coordinate grid. find point c on ab that is ⅕ of the distance from a to b. a) (-⅕, 7/4) b) (⅕, -7/4) c) (-⅕, -7/4) d) (⅕, 7/4)

Explanation:

Step1: Recall the section - formula

If a point \(C(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 1\) and \(n=4\), so the point \(C\) that is \(\frac{1}{5}\) of the distance from \(A\) to \(B\) divides the line - segment \(AB\) in the ratio \(1:4\).

Step2: Assume coordinates of \(A\) and \(B\)

Let \(A(x_1,y_1)\) and \(B(x_2,y_2)\). From the grid, assume \(A=(7,4)\) and \(B=( - 3,-4)\) (by observing the positions of \(A\) and \(B\) on the coordinate - grid).

Step3: Calculate the \(x\) - coordinate of \(C\)

Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), substitute \(m = 1\), \(n = 4\), \(x_1=7\), and \(x_2=-3\).
\[x=\frac{1\times(-3)+4\times7}{1 + 4}=\frac{-3 + 28}{5}=\frac{25}{5}=5\]

Step4: Calculate the \(y\) - coordinate of \(C\)

Using the formula \(y=\frac{my_2+ny_1}{m + n}\), substitute \(m = 1\), \(n = 4\), \(y_1 = 4\), and \(y_2=-4\).
\[y=\frac{1\times(-4)+4\times4}{1 + 4}=\frac{-4 + 16}{5}=\frac{12}{5}=\frac{7}{5}- \frac{4}{5}\]
The coordinates of \(C\) are \((5,\frac{7}{5}-4)\)

Answer:

B. \((5,\frac{7}{5}-4)\)