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Identify and graph parallel and perpendicular lines.

Learn to identify parallel and perpendicular lines, write equations, and graph them. Understand slopes for determination.

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Identify and graph parallel and perpendicular lines.

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  1. Objectives Identify and graph parallel and perpendicular lines. Write equations to describe lines parallel or perpendicular to a given line.

  2. These two lines are parallel. Parallel linesare lines in the same plane that have no points in common. In other words, they do not intersect.

  3. 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)

  4. –2x – 2x – 1 – 1 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) y + 1 = 3x – 9 3y = –2x + 8 y = 3x – 10

  5. The lines described by and represent parallel lines. They each have the slope . y = 2x – 3 y + 1 = 3(x – 3) Example 1B Continued Graph all four equations The lines described by y = 2x – 3 and y + 1 = 3(x – 3) are not parallel with any of the lines.

  6. 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

  7. 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)

  8. Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4); x = 3; y = 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) Check It Out! Example 3 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.

  9. 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. The slope is 3. y = 3x + 8 The parallel line also has a slope of 3. Step 2 Write the equation in point-slope form. Use the point-slope form. y – y1 = m(x – x1) Substitute 3 for m, 4 for x1, and 10 for y1. y – 10 = 3(x – 4)

  10. 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.

  11. The perpendicular line has a slope of because Substitute for m, –1 for y1, and 2 for x1. 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. Step 2 Write the equation in point-slope form. y – y1 = m(x – x1) Use the point-slope form.

  12. Distribute on the right side. 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. Subtract 1 from both sides.

  13. 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)

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