Parallel and Perpendicular Lines A.2E Write the equation of a line that contains a given point and is parallel to a given line. A.2F write the equation of a line that contains a given point and is perpendicular to a given line.
Slopes of Parallel Lines Words Nonvertical lines are parallel if they have the same slope and different y-intercepts. Vertical lines are parallel if they have different x-intercepts. Example: The graphs of y= 1 2 𝑥−1 and y= 1 2 𝑥−2 are lines that have the same slope, (1/2), and different y-intercepts. The lines are parallel.
Slopes of Perpendicular Lines Words: Two nonvertical lines are perpendicular if the product of their slopes is -1. A vertical line and a horizontal line are also perpendicular. Example: The graph of y= 1 2 𝑥−1 has a slope of (1/2). The graph of y=-2x + 1 has a slope of -2. Since 1 2 −2 =−1, the lines are perpendicular.
Writing an Equation of a Parallel Line A line passes through (12,5) and is parallel to the graph of 𝐲= 𝟐 𝟑 𝒙−𝟏. What equation represents the line in slope-intercept form? Start with point-slope form. y - y₁ = m(x - x₁) Substitute (12,5) for (x₁,y₁) and 𝟐 𝟑 for m. y – 5 = 𝟐 𝟑 (x – 12) Distributive Property y – 5 = 𝟐 𝟑 x - 𝟐 𝟑 (12) Simplify y – 5 = 𝟐 𝟑 x – 8 Add 5 to each side y = 𝟐 𝟑 x – 3 The graph of y = 𝟐 𝟑 x – 3 passes through (12,5) and is parallel to the graph of y = 𝟐 𝟑 x – 1
Classifying Lines Are the graphs of 4y = -5x + 12 and y = 𝟒 𝟓 x – 8 parallel, perpendicular, or neither? Step 1: Find the slope of each line by writing its equation in slope- intercept form if necessary. Only the first equation needs to be rewritten. 4y = -5x + 12 Write the first equation 𝟒𝒚 𝟒 = −𝟓𝒙+𝟏𝟐 𝟒 Divide each side by 4 Y= - 𝟓 𝟒 𝒙+𝟑 Simplify. The slope of the graph of y= - 𝟓 𝟒 𝒙+𝟑 is (-5/4) The slope of the graph of y = 𝟒 𝟓 x – 8 is (4/5) The slopes are opposite reciprocals, so the lines are perpendicular.
Writing an Equation of a Perpendicular line Which equation represents the line that passes through (2,4) and is perpendicular to the graph of y = 𝟏 𝟑 x – 1? 1. Identify the slope of the graph of the given equation. The slope is 1 3 . 2. Find the opposite reciprocal of the slope from step 1. The opposite reciprocal of 1 3 is -3. So, the perpendicular line has a slope of -3. 3. Use point-slope form to write an equation of the perpendicular line. y - y₁ = m(x - x₁) Write Point-Slope Form y – 4 = -3(x – 2 ) Substitute (2,4) for (x₁,y₁) and -3 for m. y – 4 = -3x + 6 Distributive Property y = -3x + 10 Add 4 to each side The equation is y = -3x + 10.
Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation (1,3); y=3x + 2 (2,-1); y= -x – 2 (2,-2); y=− 3 2 𝑥+6 (0,0); y= 2 3 𝑥+1 (4,2); x= -3
Determine whether the graphs of the given equations are parallel, perpendicular, or neither. y = x + 11 y – 4 = 3(x + 2) y = 𝟑 𝟒 𝒙−𝟏 y= -7 y = -x + 2 2x + 6y = 10 y = 𝟑 𝟒 𝒙+𝟐𝟗 x= 2
A path for a new city park will connect the park entrance to Main Street. The path should be perpendicular to Main Street. What is an equation that represents the path? Equation: y= 𝟓 𝟐 x – 1 Point: (0,5) y – 5 = - 𝟐 𝟓 𝐱 −𝟎 Y – 5= - 𝟐 𝟓 x y= − 𝟐 𝟓 x + 5
Equation: y= 𝟓 𝟐 x – 1 Point: (0,5) y – 5 = 𝟓 𝟐 𝐱 −𝟎 Y – 5= 𝟓 𝟐 x A bike path is being planned for the park. The bike path will be parallel to Main Street and will pass through the park entrance. What is an equation of the line that represents the bike path? Equation: y= 𝟓 𝟐 x – 1 Point: (0,5) y – 5 = 𝟓 𝟐 𝐱 −𝟎 Y – 5= 𝟓 𝟐 x Y = 𝟓 𝟐 x + 5