What is an equation of the line that passes through the point \( (4,8) \) and is perpendicular to the line \( x+2 y=12 \) ?

To find the equation of the line that passes through the point \( (4,8) \) and is perpendicular to the line \( x + 2y = 12 \), we first need to find the slope of the given line. Rearranging the equation to the slope-intercept form (y = mx + b), we get \( 2y = -x + 12 \), or \( y = -\frac{1}{2}x + 6 \). The slope of this line is \(-\frac{1}{2}\). The slope of a line perpendicular to it is the negative reciprocal, which is \(2\). Now that we have the slope of our line (2) and a point \((4,8)\), we can use point-slope form to write the equation: \[ y - 8 = 2(x - 4). \] Simplifying it gives us: \[ y - 8 = 2x - 8 \] \[ y = 2x. \] Thus, the equation of the line is \( y = 2x \). Fun fact: When two lines are perpendicular, the product of their slopes equals \(-1\). So in this case, \(-\frac{1}{2} \times 2 = -1\). If you're interested in a practical application of this concept, think about architecture and design. Engineers often need to create perpendicular support structures to evenly distribute weight. Understanding how to calculate and express these relationships through slopes and lines is crucial in ensuring structural integrity!