41) Une \( r \) has an equation of \( y+10=-2(x-6) \). Une \( s \) is perpendicular to line \( r \) and passes through \( (1,2) \). What is the equavon of line \( s \) ?

To find the equation of line \( s \) that is perpendicular to line \( r \), we first need to determine the slope of line \( r \). The given equation of Line \( r \) is \( y + 10 = -2(x - 6) \). Rearranging this into slope-intercept form (y = mx + b): 1. Distribute: \( y + 10 = -2x + 12 \) 2. Subtract 10 from both sides: \( y = -2x + 2 \) The slope of line \( r \) is \(-2\). Perpendicular lines have slopes that are negative reciprocals, so the slope of line \( s \) will be \( \frac{1}{2} \). Using the point-slope form, the equation of line \( s \) that passes through the point \( (1, 2) \) with slope \( \frac{1}{2} \) can be written as: \( y - 2 = \frac{1}{2}(x - 1) \) This can be simplified to: \( y - 2 = \frac{1}{2}x - \frac{1}{2} \) Adding 2 to both sides gives us: \( y = \frac{1}{2}x + \frac{3}{2} \) So, the equation of line \( s \) is \( y = \frac{1}{2}x + \frac{3}{2} \). This process showcases how geometric relationships between lines translate into algebraic equations, revealing the beauty of math in action! Also, remember that perpendicularity in geometry isn't confined to just two dimensions. Whew! Just imagine those right angles extending into 3D space. It's like a dance of lines, creating angles and forms. Next time you're sketching lines, think of how they interact... it's like plotting the epic story of their geometric lives!