A walking path across a park is represented by the equation $y=-4 x+10$. A new path will be built perpendicular to this path. The paths will intersect at the point $(4,-6)$. Identify the equation that represents the new path.
A. $y=\frac{1}{4} x-7$
B. $y=-\frac{1}{4} x-5$
C. $y=4 x-22$
D. $y=-4 x+10$
Answer (1)
The slope of the given path y = − 4 x + 10 is − 4 .
The slope of the new path, being perpendicular, is the negative reciprocal of − 4 , which is 4 1 .
Using the point-slope form with the point ( 4 , − 6 ) and slope 4 1 , we get y + 6 = 4 1 ( x − 4 ) .
Converting to slope-intercept form, we find the equation of the new path to be y = 4 1 x − 7 .
Explanation
Understanding the Problem The problem provides the equation of a walking path and asks us to find the equation of a new path that is perpendicular to the given path and intersects it at a specific point. We'll use the properties of perpendicular lines and the point-slope form of a linear equation to solve this.
Finding the Slope of the Existing Path The given path has the equation y = − 4 x + 10 . This is in slope-intercept form, y = m x + b , where m is the slope. Therefore, the slope of the existing path is − 4 .
Determining the Slope of the New Path Since the new path is perpendicular to the existing path, its slope is the negative reciprocal of the existing path's slope. The negative reciprocal of − 4 is 4 1 . So, the slope of the new path is 4 1 .
Using the Point-Slope Form We know the new path passes through the point ( 4 , − 6 ) . We can use the point-slope form of a linear equation, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the point and m is the slope. Plugging in the values, we get y − ( − 6 ) = 4 1 ( x − 4 ) .
Rewriting in Slope-Intercept Form Now, we rewrite the equation in slope-intercept form, y = m x + b . Starting with y + 6 = 4 1 ( x − 4 ) , we distribute the 4 1 to get y + 6 = 4 1 x − 1 . Subtracting 6 from both sides, we get y = 4 1 x − 7 .
Final Answer The equation that represents the new path is y = 4 1 x − 7 . Comparing this to the answer choices, we see that it matches option A.
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and construction. For example, when designing a building, architects need to ensure that walls are perpendicular to the ground for stability. Similarly, in road construction, engineers use perpendicular lines to design intersections and ensure safe traffic flow. The principles of perpendicularity also apply in navigation, where sailors and pilots use perpendicular bearings to determine their position.
