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)
Find the slope of the given line: y + 5 = x + 2 becomes y = x − 3 , so the slope is 1 .
Determine the slope of the perpendicular line: The negative reciprocal of 1 is − 1 .
Use the point-slope form with the point ( 2 , 5 ) and slope − 1 : y − 5 = − 1 ( x − 2 ) .
Simplify the equation: The equation of the perpendicular line is y − 5 = − ( x − 2 ) .
Explanation
Find the slope of the given line First, we need to find the slope of the given line. The equation is y + 5 = x + 2 . We can rewrite this in slope-intercept form ( y = m x + b ) to easily identify the slope.
Rewrite in slope-intercept form Subtract 5 from both sides of the equation: y = x + 2 − 5 , which simplifies to y = x − 3 . The slope of this line is m 1 = 1 .
Find the slope of the perpendicular line The slope of a line perpendicular to the given line is the negative reciprocal of the given line's slope. So, if the slope of the given line is m 1 = 1 , the slope of the perpendicular line is m 2 = − 1 1 = − 1 .
Use the point-slope form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point ( 2 , 5 ) and m is the slope of the perpendicular line, which is − 1 .
Substitute the values and simplify Substitute the values into the point-slope form: y − 5 = − 1 ( x − 2 ) . This simplifies to y − 5 = − ( x − 2 ) .
Final Answer Therefore, the equation of the line in point-slope form 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 garden and need to create a path that's perpendicular to an existing walkway. Knowing the slope of the walkway and a point where you want the new path to start, you can use the point-slope form to determine the equation of the new path. This ensures the path is perfectly perpendicular, creating an aesthetically pleasing and functional design. The point-slope form is also useful in physics, such as determining the trajectory of an object perpendicular to a certain force or direction.
