Find the equation of the line that contains the point $(6,-2)$ and is perpendicular to the line $y=-2 x+8$.
Answer (1)
Find the slope of the given line: m = − 2 .
Determine the slope of the perpendicular line: m ⊥ = 2 1 .
Use the point-slope form with the given point ( 6 , − 2 ) : y − ( − 2 ) = 2 1 ( x − 6 ) .
Simplify to slope-intercept form: y = 2 1 x − 5 .
y = 2 1 x − 5
Explanation
Problem Analysis We are given a point ( 6 , − 2 ) and a line y = − 2 x + 8 . We need to find the equation of the line that passes through the given point and is perpendicular to the given line.
Finding the Slope The slope of the given line y = − 2 x + 8 is − 2 . The slope of a line perpendicular to this line is the negative reciprocal of − 2 , which is 2 1 .
Using 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 ) = ( 6 , − 2 ) .
Substitution Substituting the values, we get y − ( − 2 ) = 2 1 ( x − 6 ) .
Simplifying the Equation Simplifying the equation, we have y + 2 = 2 1 x − 3 . Subtracting 2 from both sides, we get y = 2 1 x − 3 − 2 , which simplifies to y = 2 1 x − 5 .
Final Answer Therefore, the equation of the line is y = 2 1 x − 5 .
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 ensure structural integrity. Similarly, in urban planning, knowing how to find perpendicular lines helps in designing road layouts and ensuring efficient traffic flow.
