Question

what does it mean if the slope of a line is zero?
a. the line is at a 45-degree angle
b. the line is horizontal
c. the line is diagonal
d. the line is vertical

Explanation:

To determine the meaning of a zero - slope line, we recall the formula for the slope of a line. The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If the slope \(m = 0\), then \(y_2-y_1=0\) (since the numerator must be zero for the fraction to be zero, assuming \(x_2
eq x_1\)). This means that for all points on the line, the \(y\) - coordinate is constant. A line with a constant \(y\) - coordinate is a horizontal line.

• Option a: A line at a 45 - degree angle has a slope of 1 (since for a 45 - degree angle, the change in \(y\) equals the change in \(x\), so \(m = \frac{\Delta y}{\Delta x}=1\)), so this option is incorrect.
• Option b: As we reasoned above, a zero - slope line is horizontal, so this option is correct.
• Option c: A diagonal line has a non - zero, non - infinite slope, so this option is incorrect.
• Option d: A vertical line has an undefined slope (because the denominator \(x_2 - x_1 = 0\) in the slope formula), so this option is incorrect.

Answer:

b. The line is horizontal