Question

refer to the coordinate grid. find point x on $overline{ab}$ such that the ratio of $ax$ to $xb$ is 1:3.

Explanation:

1. First, assume the coordinates of point \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). From the grid, if we assume \(A = (- 2,3)\) and \(B=(4,3)\).

• The formula to find the coordinates of a point \(X=(x,y)\) that divides the line - segment \(\overline{AB}\) 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, \(m = 1\) and \(n = 3\), \(x_1=-2\), \(x_2 = 4\), \(y_1 = 3\), \(y_2 = 3\).

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

• Substitute the values into the \(x\) - coordinate formula:
• \(x=\frac{1\times4+3\times(-2)}{1 + 3}=\frac{4-6}{4}=\frac{-2}{4}=-\frac{1}{2}\).
• Calculate the \(y\) - coordinate of point \(X\):
• Substitute the values into the \(y\) - coordinate formula: \(y=\frac{1\times3+3\times3}{1 + 3}=\frac{3 + 9}{4}=\frac{12}{4}=3\).

Answer:

The coordinates of point \(X\) are \((-\frac{1}{2},3)\)