What is the equation of a line that is perpendicular to $y=2 x+4$ and passes through the point $(4,6)$?
A. $y=-\frac{1}{2} x+6$
B. $y=\frac{1}{2} x+4$
C. $y=-\frac{1}{2} x+4$
D. $y=-\frac{1}{2} x+8$
Answer (1)
Determine the slope of the given line: The slope of y = 2 x + 4 is 2 .
Calculate the slope of the perpendicular line: The negative reciprocal of 2 is − 2 1 .
Use the point-slope form with the point ( 4 , 6 ) and slope − 2 1 : y − 6 = − 2 1 ( x − 4 ) .
Convert to slope-intercept form: y = − 2 1 x + 8 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to the line y = 2 x + 4 and passes through the point ( 4 , 6 ) .
Finding the Slope of the Given Line First, we need to find the slope of the given line. The equation y = 2 x + 4 is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In this case, the slope of the given line is 2 .
Finding the Slope of the Perpendicular Line Next, we need to find the slope of the line perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original line's slope. So, the slope of the perpendicular line is − 2 1 .
Using the Point-Slope Form Now, we use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point. We have the slope m = − 2 1 and the point ( 4 , 6 ) . Plugging these values into the point-slope form, we get:
y − 6 = − 2 1 ( x − 4 )
Rewriting in Slope-Intercept Form Now, we rewrite the equation in slope-intercept form, y = m x + b .
y − 6 = − 2 1 x + 2
y = − 2 1 x + 2 + 6
y = − 2 1 x + 8
Final Answer Therefore, the equation of the line that is perpendicular to y = 2 x + 4 and passes through the point ( 4 , 6 ) is y = − 2 1 x + 8 .
Examples
Understanding perpendicular lines is crucial in architecture and construction. For example, when designing a building, ensuring walls are perpendicular to the ground is essential for stability. The equation of a line helps architects calculate angles and slopes to create precise and safe structures. Similarly, in urban planning, knowing how streets intersect at right angles can optimize traffic flow and improve overall city layout.
