Question

what is the equation of the line that is perpendicular to the given line and passes through the point $(2,6)$?
$x = 2$
$x = 6$
$y = 2$
$y = 6$

Explanation:

Step1: Determine the slope of the given line

The given line passes through \((-8,-4)\) and \((8,-4)\). Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), we have \(m=\frac{-4-(-4)}{8 - (-8)}=\frac{0}{16}=0\). So, the given line is a horizontal line (\(y=-4\)).

Step2: Find the slope of the perpendicular line

A line perpendicular to a horizontal line is a vertical line. The slope of a vertical line is undefined.

Step3: Use the point \((2,6)\) to write the equation

Since the perpendicular line is vertical and passes through \((2,6)\), its equation is \(x = 2\) (a vertical line has the form \(x = k\), where \(k\) is the \(x\)-coordinate of any point on the line).

Answer:

\(x = 2\) (First option)