Question

a line passes through the two given points. is it vertical, horizontal, or neither? (1, 0), (0, 0) neither horizontal vertical

Explanation:

Step1: Recall the properties of horizontal and vertical lines

A horizontal line has the equation \(y = k\) (constant \(y -\)value for all points on the line), so for any two points \((x_1,y_1)\) and \((x_2,y_2)\) on a horizontal line \(y_1=y_2\). A vertical line has the equation \(x = k\) (constant \(x -\)value for all points on the line), so for any two points \((x_1,y_1)\) and \((x_2,y_2)\) on a vertical line \(x_1=x_2\).

Step2: Check the \(x\) and \(y\) values of the given points

For the points \((1,0)\) and \((0,0)\), the \(y -\)values are the same (\(y = 0\) for both points). But a horizontal line has the form \(y=0\) (the \(x -\)axis). However, let's also check the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((x_1,y_1)=(1,0)\) and \((x_2,y_2)=(0,0)\), we have \(m=\frac{0 - 0}{0 - 1}=0\) (slope of a horizontal line is \(0\)). But wait, a vertical line has an undefined slope (\(x_2=x_1\)). Here \(x_1
eq x_2\) (\(1
eq0\)). A non - vertical and non - horizontal line has a non - zero and defined slope. But in this case, although the slope \(m = 0\) (horizontal line property in terms of slope), but if we consider the general form: a horizontal line is parallel to the \(x -\)axis and has all \(y\) values equal. A vertical line is parallel to the \(y -\)axis and has all \(x\) values equal. The line passing through \((1,0)\) and \((0,0)\) is the \(x -\)axis (\(y = 0\)), which is a horizontal line.

Answer:

Horizontal