Objectives Identify and graph parallel and perpendicular lines. Write equations to describe lines parallel or perpendicular to a given line.
These two lines are parallel These two lines are parallel. Parallel lines are lines in the same plane that have no points in common. In other words, they do not intersect.
Identify which pairs of lines are parallel. (Must have the exact same slope) 1. y = ⅔ x + 5 and y = -⅔ x + 1 2. y = -6x + 4 and y = -6x - 3 3. y = ⅘ x – 2 and y = ⅘ x + 6 Write an equation parallel to y = 2x – 1 (Must have the exact same slope as this equation)
Example 1B Continued Identify which lines are parallel. y = 2x – 3 y = -⅔ x + 3 2x + 3y = 8 y + 1 = 3(x – 3) Make sure all equations are in slope-intercept form to determine the slope. 2x + 3y = 8 y + 1 = 3(x – 3) –2x – 2x y + 1 = 3x – 9 3y = –2x + 8 – 1 – 1 y = 3x – 10
Graph all four equations Example 1B Continued Graph all four equations y = 2x – 3 y + 1 = 3(x – 3) The lines described by y = 2x – 3 and y + 1 = 3(x – 3) are not parallel with any of the lines. The lines described by and represent parallel lines. They each have the slope .
Perpendicular lines are lines that intersect to form right angles (90°). Perpendicular lines have opposite reciprocal slopes. Flip the fraction Change the sign Example: -3/5 and 5/3 Example: 2/1 and -1/2
Identify which pairs of lines are perpendicular. (Must have opposite reciprocal slopes) 1. y = ⅔ x + 5 and y = -⅔ x + 6 2. y = -6x + 4 and y = ⅙ x + 1 3. y = -x – 2 and y = x - 3 Write an equation perpendicular to y = -¾x + 1 (Must have an opposite reciprocal slope to this one)
Check It Out! Example 3 Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y = The graph described by x = 3 is a vertical line, and the graph described by y = –4 is a horizontal line. These lines are perpendicular. x = 3 The slope of the line described by y – 6 = 5(x + 4) is 5. The slope of the line described by y = is y = –4 y – 6 = 5(x + 4)
Example 5A: Writing Equations of Parallel and Perpendicular Lines Write an equation in slope-intercept form for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8. Step 1 Find the slope of the line. y = 3x + 8 The slope is 3. The parallel line also has a slope of 3. Step 2 Write the equation in point-slope form. y – y1 = m(x – x1) Use the point-slope form. Substitute 3 for m, 4 for x1, and 10 for y1. y – 10 = 3(x – 4)
Example 5A Continued Write an equation in slope-intercept form for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8. Step 3 Write the equation in slope-intercept form. y – 10 = 3(x – 4) y – 10 = 3x – 12 Distribute 3 on the right side. y = 3x – 2 Add 10 to both sides.
Example 5B: Writing Equations of Parallel and Perpendicular Lines Write an equation in slope-intercept form for the line that passes through (2, –1) and is perpendicular to the line described by y = 2x – 5. Step 1 Find the slope of the line. y = 2x – 5 The slope is 2. The perpendicular line has a slope of because Step 2 Write the equation in point-slope form. y – y1 = m(x – x1) Use the point-slope form. Substitute for m, –1 for y1, and 2 for x1.
Example 5B Continued Write an equation in slope-intercept form for the line that passes through (2, –1) and is perpendicular to the line described by y = 2x – 5. Step 3 Write the equation in slope-intercept form. Distribute on the right side. Subtract 1 from both sides.
Determine if the two equations are perpendicular Hint: make sure each looks like y = mx + b first 2y – 8 = 3x and y = ⅔ x + 1 Write an equation for the line that goes through the point (2, 1) and is parallel to y = -3x + 2. Hint: Plug m, x, and y into (y – y1)=m(x – x1)