Find the equation of the line that is perpendicular to [tex]$y=-\frac{2}{3} x$[/tex] and contains the point (4,-8).
[tex]$y=\frac{[?]}{[\square]} x+[]$[/tex]
Answer (1)
Find the slope of the perpendicular line: m = 2 3 .
Use the point-slope form: y − ( − 8 ) = 2 3 ( x − 4 ) .
Simplify the equation: y + 8 = 2 3 x − 6 .
Solve for y : y = 2 3 x − 14 . The equation of the line is y = 2 3 x − 14 .
Explanation
Find the slope of the perpendicular line First, we need to find the slope of the line that is perpendicular to the given line. The given line is y = − 3 2 x . The slope of this line is − 3 2 . The slope of a line perpendicular to this line is the negative reciprocal of − 3 2 , which is 2 3 .
Use the point-slope form Next, 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 this case, m = 2 3 and ( x 1 , y 1 ) = ( 4 , − 8 ) .
Substitute the values Substitute the values into the point-slope form: y − ( − 8 ) = 2 3 ( x − 4 ) .
Simplify the equation Simplify the equation to slope-intercept form, y = m x + b . First, we have y + 8 = 2 3 x − 2 3 ( 4 ) .
Continue simplifying Then, y + 8 = 2 3 x − 6 .
Isolate y Subtract 8 from both sides: y = 2 3 x − 6 − 8 .
Final equation Simplify: y = 2 3 x − 14 .
Examples
Understanding perpendicular lines is crucial in many real-world applications, such as architecture and construction. For example, when building a house, the walls need to be perpendicular to the ground to ensure stability. The equation of a line perpendicular to another line can help architects and engineers design structures that are safe and sound. In computer graphics, perpendicular lines are used to create realistic images and animations. By understanding the relationship between slopes and perpendicularity, we can solve various practical problems in different fields.
