What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point $(2,5)$?
A. $y+5=x+2$
B. $y-2=x-5$
C. $y-5=-(x-2)$
D. $y+2=-(x+5)$
Answer (1)
Rewrite the given equation y + 5 = x + 2 in slope-intercept form to find its slope, which is 1 .
Determine the slope of the perpendicular line by taking the negative reciprocal of the given line's slope, resulting in a slope of − 1 .
Use the point-slope form y − y 1 = m ( x − x 1 ) with the point ( 2 , 5 ) and the perpendicular slope − 1 to create the equation.
The equation of the perpendicular line in point-slope form is y − 5 = − ( x − 2 ) .
Explanation
Understanding the Problem We are given a line and a point, and we want to find the equation of the line that is perpendicular to the given line and passes through the given point. The equation should be in point-slope form.
Finding the Slope of the Given Line The given line is y + 5 = x + 2 . We can rewrite this in slope-intercept form by subtracting 5 from both sides: y = x − 3 . The slope of this line is m 1 = 1 .
Finding the Slope of the Perpendicular Line The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. So, the slope of the perpendicular line is m 2 = − m 1 1 = − 1 1 = − 1 .
Using the Point-Slope Form We are given the point ( 2 , 5 ) that the perpendicular line passes through. The point-slope form of a line is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope.
Writing the Equation of the Perpendicular Line Plugging in the point ( 2 , 5 ) and the slope m 2 = − 1 , we get the equation of the perpendicular line in point-slope form: y − 5 = − 1 ( x − 2 ) , which simplifies to y − 5 = − ( x − 2 ) .
Final Answer Therefore, the equation of the line that is perpendicular to the given line and passes through the point ( 2 , 5 ) is y − 5 = − ( x − 2 ) .
Examples
Imagine you're designing a rectangular garden and need to ensure the paths are perfectly perpendicular to the edges. Knowing how to find the equation of a perpendicular line helps you calculate the exact layout and angles needed for the paths to meet the garden edges at a 90-degree angle, creating a neat and organized space. This concept is also crucial in architecture and construction for ensuring structures are aligned correctly.
