Parallel and Perpendicular Lines By Lindsay Hojnowski (2014) Buffalo State College 04/2014L. Hojnowski © 20141 Click here to play tutorial introduction.

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Presentation transcript:

Parallel and Perpendicular Lines By Lindsay Hojnowski (2014) Buffalo State College 04/2014L. Hojnowski © Click here to play tutorial introduction Parallel Lines Perpendicular Lines

Learning Objectives The learner will be able to put the equations in slope-intercept form to identify the slope 85% of the time. The learner will be able to identify what the parallel or perpendicular slope is 85% of the time. Student wills be able to use the point-slope formula to find parallel lines given a point and a line (using the given slope) 80% of the time. Students will be able to use the point-slope formula to find perpendicular lines given a point and a line (using the given slope) 80% of the time. 04/2014L. Hojnowski © Aim for the Target

Menu 04/2014L. Hojnowski © References Characteristics of Parallel Lines Parallel Lines- Steps Given a point and an equation Parallel Lines- Example 1 Perpendicular Lines- Example 1 Parallel Lines- Example 3 Characteristics of Perpendicular Lines Perpendicular Lines- Steps Given a point and an equation Parallel Lines- Example 2 Perpendicular Lines- Example 2 Perpendicular Lines- Example 3 Determine whether parallel, perpendicular, or neither- Steps Determine- Example 1 Determine- Example 2 Determine- Example 3 Quiz Question #1 Quiz Question #2 Quiz Question #7 Quiz Question #6 Quiz Question #5 Quiz Question #4 Quiz Question #3

Characteristics of Parallel Lines Parallel lines: 1)Are lines that do not intersect 2)Have different y-intercepts - Click on the picture below to see a video on how to write a parallel line to another line using point-slope form 04/2014L. Hojnowski © Parallel Lines- JMAP Video

Parallel Lines- Steps Given a point and an equation 04/2014L. Hojnowski © Steps to writing a parallel line STEPS: 1)Rewrite the given equation into slope- intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the parallel slope (found in step 1) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

Parallel Lines- Example 1 04/2014L. Hojnowski © Example 1: Write an equation in slope-intercept form for the line that passes through (-2, 2) and is parallel to y = 4x – 2. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = 43) y – 2 = 4 (x + 2) 2)y – y 1 = m (x – x 1 ) y – 2 = 4x + 8 y – 2 = 4 (x - - 2) y – 2 = 4 (x + 2) 4) y = 4x + 10 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the parallel slope (found in step 1) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

Parallel Lines- Example 2 04/2014L. Hojnowski © Example 2: Write an equation in slope-intercept form for the line that passes through (6, 4) and is parallel to y = (1/3)x + 1. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = 1/33) y – 4 = (1/3) (x - 6) 2)y – y 1 = m (x – x 1 ) y – 4 = (1/3)x - 2 y – 4 = (1/3) (x - 6) ) y = (1/3)x + 2 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the parallel slope (found in step 1) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

Parallel Lines- Example 3 04/2014L. Hojnowski © Example 3: Write an equation in slope-intercept form for the line that passes through (-1, 6) and is parallel to 3x + y = 12. **Use the point-slope formula** The equation is NOT in slope-intercept form, m = ? 3) y – 6 = -3 (x – -1) **In order to identify the slope, solve for y! y – 6 = -3 (x + 1) 3x + y = 12 y – 6 = -3x x y = -3x +12 m = -3 4) y – 6 = -3x ) y – y 1 = m (x – x 1 ) y = -3x +3 y – 6 = -3 (x – -1) Example of a given point and a line

Characteristics of Perpendicular Lines 04/2014L. Hojnowski © Perpendicular Lines: 1)Are lines that intersect at right angles 2)Have negative reciprocal slopes -Example: m = 2  m = -1/2 - Click on the picture below to see a video to review how to write a perpendicular line to another line using slope-intercept form (you can use point-slope formula just like parallel lines) Perpendicular Lines- JMAP Video

Perpendicular Lines- Steps Given a point and an equation 04/2014L. Hojnowski © Steps to writing a perpendicular l line STEPS: 1)Rewrite the given equation into slope- intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the perpendicular slope (negative reciprocal) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

Perpendicular Lines- Example 1 04/2014L. Hojnowski © Example 1: Write an equation in slope-intercept form for the line that passes through (4, 2) and is perpendicular to y = (1/2)x + 1. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = (1/2)3) y – 2 = -2 (x - 4) Perpendicular slope: -2 y – 2 = -2x + 8 2) y – y 1 = m (x – x 1 ) y – 2 = -2 (x - 4) 4) y= -2x + 10 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the perpendicular slope (negative reciprocal) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

Perpendicular Lines- Example 2 04/2014L. Hojnowski © Example 2: Write an equation in slope-intercept form for the line that passes through (-5, -1) and is perpendicular to y = (5/2)x - 3. **Use the point-slope formula** 1)The equation is in slope-intercept form, m = (5/2)3) y + 1= (-2/5) (x + 5) Perpendicular slope: (-2/5) y + 1= (-2/5)x - 2 2)y – y 1 = m (x – x 1 ) y – -1 = (-2/5) (x - - 5) 4) y= (-2/5)x - 3 STEPS: 1)Rewrite the given equation into slope-intercept from (y = mx + b), if necessary, and identify the slope (m) 2)Plug in the given point and the perpendicular slope (negative reciprocal) in the point-slope formula (y – y 1 = m (x – x 1 ) 3)Distribute and simplify (if necessary) 4)Solve for y

Example 3: Write an equation in slope-intercept form for the line that passes through (-4, 6) and is perpendicular to 2x + 3y = 12. **Use the point-slope formula** 1) The equation is NOT in slope-intercept form, m = ? **In order to identify the slope, solve for y! 2x + 3y = 12 2) y – y 1 = m (x – x 1 ) 3) y – 6 = (3/2)(x + 4) -2x -2x y – 6 = (3/2)(x – -4) y – 6 = (3/2)x + 6 3y = -2x y = (-2/3)x + 4 4) y = (3/2)x + 12 m = -2/3 Perpendicular slope: (3/2) Perpendicular Lines- Example 3 04/2014L. Hojnowski © Given a Point and a Line

Determine whether parallel, perpendicular, or neither- Steps 04/2014L. Hojnowski © STEPS: 1)Rewrite both equation into slope-intercept form (y = mx + b) and identify each slope 2)Compare the slopes to see if they are the same, negative reciprocal, or neither Example of Parallel Lines- Same Slope Example of Perpendicular l Lines- Negative Reciprocal Slope Example of Neither Parallel or Perpendicular Lines

Determine whether parallel, perpendicular, or neither- Example 1 04/2014L. Hojnowski © Example1: Determine whether the graphs of the pair of equations are parallel, perpendicular, or neither. 3x + 5y = 105x – 3y = -6 STEPS: 1)Rewrite both equation into slope-intercept form (y = mx + b) and identify each slope 2)Compare the slopes to see if they are the same, negative reciprocal, or neither 5x – 3y = -6 -5x -3y = -5x y = (-5/-3)x + 2 m = (5/3) 3x + 5y = 10 -3x 5y = -3x y = (-3/5)x + 2 m = (-3/5) PERPENDICULAR- they have negative reciprocal slopes

Determine whether parallel, perpendicular, or neither- Example 2 04/2014L. Hojnowski © Example 2: Determine whether the graphs of the pair of equations are parallel, perpendicular, or neither. 2x - 8y = -24 x – 4y = 4 STEPS: 1)Rewrite both equation into slope-intercept form (y = mx + b) and identify each slope 2)Compare the slopes to see if they are the same, negative reciprocal, or neither PARALLEL- they have the same slope x – 4y = 4 -x -4y = -x y = (-1/-4)x - 1 m = (1/4) 2x - 8y = x -8y = -2x y = (-2/-8)x + 3 m = (1/4)

Determine whether parallel, perpendicular, or neither- Example 3 04/2014L. Hojnowski © Example 3: Determine whether the graphs of the pair of equations are parallel, perpendicular, or neither. -3x + 4y = 8 -4x + 3y = -6 NEITHER- they aren’t the same slope and are not negative reciprocals They are reciprocals but not negative reciprocals -4x + 3y = -6 +4x 3y = 4x y = (4/3)x - 2 m = (4/3) -3x + 4y = 8 +3x 4y = 3x y = (3/4)x + 2 m = (3/4)

Quiz Question #1 04/2014L. Hojnowski © What is the perpendicular slope of the line that passes through the line: y = (-3/4)x + 4 ? a. a. -4/3 b. 3/4 c. 4/3d. -3/4 b. c. d.

Try Again… 04/2014L. Hojnowski © This slope is the reciprocal of the slope given. Perpendicular slopes are the negative reciprocals. Quiz Question #1 Quiz Question #2 Try Again Perpendicular Lines

Try Again… 04/2014L. Hojnowski © Quiz Question #1 Quiz Question #2 Try Again This slope is the negative of the slope given. Perpendicular slopes are the negative reciprocals. Intersecting Lines

Correct!! 04/2014L. Hojnowski © Quiz Question #1 Quiz Question #2 Smile You are correct! Perpendicular slopes are the negative reciprocals. Negative Reciprocal Slope

Try Again… 04/2014L. Hojnowski © Quiz Question #1 Quiz Question #2 Try Again This slope is the same as the slope given. This would be correct if the question asked for the parallel slope. Perpendicular slopes are the negative reciprocals. Parallel Lines

Quiz Question # 2 04/2014L. Hojnowski © Which line is parallel to the line 4x + y = 3? a.a. y = (1/4) x – 1 b. y = 4x + 2 c. y = (-1/4) x – 6 d. y = -4x + 5b. c. d.

Try Again… 04/2014L. Hojnowski © If the question asked you to write the equation of the line in point-slope form, you would be correct. From this answer, you need to distribute and get y by itself. Quiz Question #1 Quiz Question #2 Try Again Quiz Question #3

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 If the question asked you to write the equation of the line in standard, you would be correct. You went one step too far.

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 There is a sign off in your equation. This is in slope-intercept form though! You are right about that.

Correct!! 04/2014L. Hojnowski © You plugged in and solved correctly! Slope-intercept form is y = mx + b; by just knowing this detail, answers #1 and #3 could be eliminated. Smile Quiz Question #1 Quiz Question #2 Quiz Question #3

04/2014L. Hojnowski © What is the slope of the line 2x + 7y = -35? a.a. 2/7b. -2/7c. 7/2 d. -7/2b. c. d.

Try Again… 04/2014L. Hojnowski © When you start to solve for y, you subtract 2x from both sides. This leads the slope to be negative and not positive. Quiz Question #2 Quiz Question #1 Try Again Quiz Question #4 Quiz Question #3

Correct!! 04/2014L. Hojnowski © Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 You solved correctly! The slope is -2/7! When solving for y, you subtracted 2x from both sides and divided by 7.

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 This is the perpendicular slope of the equation you get after you solve. If the question asked for the perpendicular slope, you would be correct.

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 This is the reciprocal slope of the equation you get after you solve.

Quiz Question # 4 04/2014L. Hojnowski © Write an equation in slope-intercept form for the line that passes through (0, 4) and is parallel to y = -4x + 5. a. a. y = -4xb. y = -4x - 4c. y = (1/4)x + 4d. y = -4x + 4b. c. d.

Try Again… 04/2014L. Hojnowski © Careful anything multiplied by zero is zero! Try again using this knowledge. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Careful with your signs and solving! To get the -4 to the other side, you must add 4 to both sides. You don’t subtract. Try again using this knowledge.

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 This would be correct if the question asked for the perpendicular line. Remember parallel lines have the same slope. Try again using this knowledge.

Correct!! 04/2014L. Hojnowski © Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 You plugged in and solved correctly! You found the parallel slope.

Quiz Question # 5 04/2014L. Hojnowski © Write an equation in slope-intercept form for the line that passes through (-8, 0) and is perpendicular to y = (-1/2)x - 4 a.a. y = (-1/2)x - 4b. y = 2x + 16c. y = -2x - 16 d. y = (1/2)x + 4b. c. d.

Try Again… 04/2014L. Hojnowski © Careful the questions asked for a line that was perpendicular to the given line. This equations had a parallel slope. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6

Correct!! 04/2014L. Hojnowski © Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 You plugged in and solved correctly! Awesome job plugging in the negative reciprocal slope!

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 The slope in the equation is the reciprocal of the slope given in the problem. Perpendicular slopes are the negative reciprocals.

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 The slope in the equation is the negative of the slope given in the problem. Perpendicular slopes are the negative reciprocals. Also, be careful of your signs when multiplying.

Quiz Question # 6 6. Determine whether 2x + 7y = -35 and 4x + 14y = -42 are parallel, perpendicular, or neither. a.a. neither b. parallel c. perpendicular b. c. 04/2014L. Hojnowski ©

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Take your time and try to find your mistake.

Correct!! 04/2014L. Hojnowski © Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 You solved both of the equations correctly for y!

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Perpendicular lines have negative reciprocal slopes.

Quiz Question # 7 6. Determine whether 3x + 5y = 10 and 5x – 3y= -6 are parallel, perpendicular, or neither. a.a. neither b. parallel c. perpendicular b. c. 04/2014L. Hojnowski ©

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Take your time and try to find your mistake.

Try Again… 04/2014L. Hojnowski © Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Be careful of your signs when you are solving. Parallel lines have the same slope.

Correct!! 04/2014L. Hojnowski © Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 You solved both of the equations correctly for y!

References McGraw-Hill Companies. (2014). Glencoe Algebra 1 Common Core Edition. New York: McGraw Hill. Seminars.usb.ac.ir. (2011). Hitting the objectives, Retrieved on September 14 th, 2012, from facilitator. facilitator Smiley Face, Retrieved on September 14 th, 2012, from ges/happy-face1.png. ges/happy-face1.png Wee, E. (2011). Try again, Retrieved on September 15 th, 2012, from 3-caring-for-children.html. 3-caring-for-children.html 04/2014L. Hojnowski © Reference from the dictionary