Which equation is perpendicular to [tex]$y=4 x-12$[/tex]?
[tex]$y=-4 x-2$[/tex] [tex]$y=\frac{1}{4} x-10$[/tex]
[tex]$y=-\frac{1}{4} x+5$[/tex] [tex]$y=4 x+11$[/tex]
Answer (1)
The problem asks to find the equation of a line perpendicular to y = 4 x − 12 . The slope of the given line is 4. A perpendicular line has a slope that is the negative reciprocal of 4, which is − 4 1 . Among the options, y = − 4 1 x + 5 has the correct slope. Therefore, the answer is y = − 4 1 x + 5 . The key steps are:
Identify the slope of the given line: m = 4 .
Calculate the negative reciprocal: − 4 1 .
Find the equation with the matching slope: y = − 4 1 x + 5 .
The equation of the perpendicular line is: y = − 4 1 x + 5 .
Explanation
Understanding the Problem We are given the equation of a line: y = 4 x − 12 . We need to find which of the given options is perpendicular to this line.
Perpendicular Lines Two lines are perpendicular if the product of their slopes is -1. The slope of the given line y = 4 x − 12 is 4.
Finding the Slope We need to find a line with a slope m such that 4 × m = − 1 . Solving for m , we get m = − 4 1 .
Checking the Options Now we examine the given options to find the line with a slope of − 4 1 :
y = − 4 x − 2 has a slope of -4.
y = 4 1 x − 10 has a slope of 4 1 .
y = − 4 1 x + 5 has a slope of − 4 1 .
y = 4 x + 11 has a slope of 4.
Final Answer The line y = − 4 1 x + 5 has a slope of − 4 1 , which is the slope we are looking for. Therefore, this line is perpendicular to y = 4 x − 12 .
Examples
Understanding perpendicular lines is crucial in various real-world applications. For instance, architects and engineers use this concept to design buildings and structures that are stable and aligned correctly. Imagine designing a bridge where the support beams need to be perfectly perpendicular to the road surface to ensure maximum strength and safety. Similarly, in navigation, understanding perpendicular paths helps ships and airplanes maintain safe distances and avoid collisions. This principle ensures precision and safety in both design and movement.
