Find the equation of the line that is perpendicular to $y=-2 x+1$ and contains the point $(8,2)$.
$y=\frac{[?]}{[1]} x+[]$
Answer (1)
Determine the slope of the given line: The slope of y = − 2 x + 1 is − 2 .
Calculate the slope of the perpendicular line: The slope is 2 1 .
Use the point-slope form with the point ( 8 , 2 ) : y − 2 = 2 1 ( x − 8 ) .
Convert to slope-intercept form: y = 2 1 x − 2 . The final answer is y = 2 1 x − 2 .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to a given line and passes through a specific point.
Identifying the Given Information The given line is y = − 2 x + 1 . We need to find a line perpendicular to this one that passes through the point ( 8 , 2 ) .
Finding the Slope of the Perpendicular Line The slope of the given line is − 2 . The slope of a line perpendicular to this line is the negative reciprocal of − 2 , which is 2 1 .
Using the Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is the given point. In our case, m = 2 1 and ( x 1 , y 1 ) = ( 8 , 2 ) . Plugging these values in, we get y − 2 = 2 1 ( x − 8 ) .
Converting to Slope-Intercept Form Next, we convert the equation to slope-intercept form, which is y = m x + b . Starting with y − 2 = 2 1 ( x − 8 ) , we distribute the 2 1 to get y − 2 = 2 1 x − 4 . Adding 2 to both sides, we get y = 2 1 x − 2 .
Stating the Final Answer Therefore, the equation of the line is y = 2 1 x − 2 .
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 to maintain structural integrity. Similarly, in road construction, engineers use perpendicular lines to design intersections and ensure safe traffic flow. This problem demonstrates a fundamental concept in geometry that has practical implications in various fields.
