What is an equation of the line that passes through the point \( (6,-4) \) and is perpendicular to the line \( 2 x-y=4 \) ?

To find the equation of a line that is perpendicular to the line \( 2x - y = 4 \), we first need to find the slope of the given line. Rewriting it in slope-intercept form \( y = mx + b \), we get \( y = 2x - 4 \), indicating that the slope (\( m \)) is \( 2 \). The slope of a line that is perpendicular to this would be the negative reciprocal, which is \( -\frac{1}{2} \). Using the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \((x_1, y_1) = (6, -4)\) and \( m = -\frac{1}{2} \), we plug in these values: \[ y - (-4) = -\frac{1}{2}(x - 6). \] This simplifies to: \[ y + 4 = -\frac{1}{2}x + 3, \] resulting in: \[ y = -\frac{1}{2}x - 1. \] This is the equation of the required line! Now, here’s a bit of fun history: Did you know that the concept of perpendicular lines has its roots in the work of ancient Greek mathematicians? Euclid’s "Elements" laid down the foundational rules of geometry, establishing the relationship between angles and lines. Moreover, perpendicular lines aren’t just a mathematical curiosity; they’re everywhere in real life! For example, think about road designs or the way buildings are constructed. Architects and engineers use perpendicular lines to create stable and efficient structures, ensuring that everything stands straight and sturdy!