Parallel & Perpendicular Lines 2010 Preston PowerPoints Real World Application How do you get the ball in the hole? 2010 Preston PowerPoints Lesson Objectives 1) Identify lines that are parallel and perpendicular. 2) Find parallel and perpendicular slopes of linear equations. 3) Write linear equations of lines that are parallel and perpendicular using a point, y-intercept and horizontal and vertical lines. 2010 Preston PowerPoints Review Opposite Operation

3x = 9 +3 +3 Dont add 3 to both sides. 2010 Preston PowerPoints Review Opposite Operation 3x = 9 3 3 Divide by 3 on both sides. 2010 Preston PowerPoints Definition Parallel Lines ( ) 2 different lines that have slope the same slope. slope (m) = 3 slope (m) = 4 slope = 3 slope = 4 2010 Preston PowerPoints

Example 1) Are these lines Parallel? 5 4 3 2 -2 -5 -4 -3 -2 -1 -2 1 -1 +2 1 -2 -3

-4 -5 +2 2 3 4 5 2 2 = 1 2 2 =

1 Yes 2010 Preston PowerPoints Example 2) Write an equation in slopeintercept form of a line parallel to y = x + 6 with a 4 y-intercept of 4. y= 1 1x + 6 Same m +b y = mx Slope y = 1x + b y = 1x 4 2010 Preston PowerPoints Whiteboard Practice 3) Write an equation in slopeintercept form of a line

parallel to y = 2x + 1 with a 6 y-intercept of 6. y = 2x 2 +1 Same m +b y = mx Slope y = 2x + b y = 2x 6 2010 Preston PowerPoints Whiteboard Practice 4) Write an equation in slopeintercept form of a line parallel to y = 3x 7 with a 8 y-intercept of 8. y = 3x 3 7 Same m +b y = mx Slope

y = 3x + b y = 3x + 8 2010 Preston PowerPoints Real World Application Knowing the how the ball will come off the wall (perpendicular angle) will help you sink the ball. 2010 Preston PowerPoints Real World Application Knowing the how the ball will come off the wall (perpendicular angle) will help you sink the ball. 2010 Preston PowerPoints Real World Application Knowing the how the ball will come off the wall (perpendicular angle) will help you sink the ball.

2010 Preston PowerPoints Definition/Example Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m Opposite 5) slope (m) = 3 Reciprocal slope = 3 3 1 1 2010 Preston PowerPoints Definition/Example Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m Opposite 5) slope (m) = 3 Reciprocal slope = 3

1 3 1 2010 Preston PowerPoints Whiteboard Practice 6) Line b has a slope (m) = 2 What is the perpendicular slope of line b? slope = 2 + 2 Opposite Reciprocal 1 1 Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Whiteboard Practice 6) Line b has a slope (m) = 2 What is the perpendicular slope

of line b? slope = 2 1 Opposite Reciprocal 1 +2 Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Example 7) Line f has a slope (m) = 7 What is the perpendicular slope of line f ? slope = 7 + 7 Opposite Reciprocal 1 1 Perpendicular Lines ( ) - one line has a slope of m and the

1 other line has a slope of m 2010 Preston PowerPoints Example 7) Line f has a slope (m) = 7 What is the perpendicular slope of line f ? slope = 7 1 Opposite Reciprocal 1 +7 Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Example 5

2 8) Line k has a slope (m) = What is the perpendicular slope of line k ? 5 slope = Opposite 2 Reciprocal Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Example 5 2 8) Line k has a slope (m) = What is the perpendicular slope

of line k ? 2 slope = Opposite 5 Reciprocal Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Example 3 2 22 3 9) Line p has a slope of and line r has a slope of Are the two lines parallel, perpendicular or neither? 3

Same Slope?No slope = 2 Perpendicular 2 Slope? Yes slope = 3 3 2 Opposite = Reciprocal 2 3 2010 Preston PowerPoints Whiteboard Practice 7 3 10) Line j has a slope (m) = What is the perpendicular slope

of line j ? 7 slope = Opposite 3 Reciprocal Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Whiteboard Practice 7 3 10) Line j has a slope (m) = What is the perpendicular slope of line j ? 3

slope = Opposite 7 Reciprocal Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Whiteboard Practice 1 6 11 6 11) Line t has a slope of and line p has a slope of Are the two lines parallel, perpendicular or neither? 1 Same Slope?No slope =

6 Perpendicular 1 Slope? No slope = 6 1 6 Opposite = Reciprocal 6 1 2010 Preston PowerPoints Whiteboard Practice 12) Line w has a slope (m) = 5 What is the perpendicular slope of line w ? slope = 5 + 5 Opposite Reciprocal 1 1

Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Example 12) Line w has a slope (m) = 5 What is the perpendicular slope of line w ? slope = 5 1 Opposite Reciprocal 1 +5 Perpendicular Lines ( ) - one line has a slope of m and the 1 other line has a slope of m 2010 Preston PowerPoints Whiteboard Practice

13) Are these lines Perpendicular? Yes 5 4 +2 3 2 +3 -5 -4 -3 1 -2 -1 -2 -1 1

-2 -3 -4 -5 +3 2 3 4 5 3 2 3 = 2 2 3

3 = 2 2010 Preston PowerPoints Example 14) Write an equation in slope-intercept form of a line that has a y-intercept of 5 & that is perpendicular to the 3 + 4. 4 line y = 3x slope (m) = 3 y = m mx + b slope = 3 + 3 1 1 Opposite Reciprocal 2010 Preston PowerPoints Example

14) Write an equation in slope-intercept form of a line that has a y-intercept of 5 & that is perpendicular to the line y = 3x + 4. slope (m) = 3 y = m mx + b 1 slope = 3 +3 1 1 y= x + b5 3 2010 Preston PowerPoints Example 14) Write an equation in slope-intercept form of a line that has a y-intercept of 5 & that is perpendicular to the line y = 3x + 4.

slope (m) = 3 y = m mx + b 1 slope = 3 +3 1 1 y= x5 3 2010 Preston PowerPoints Whiteboard Practice 15) Write an equation in slope-intercept form of a line that has a y-intercept of 3 & that is perpendicular to the 7 2. 2 line y = 7x slope (m) = 7 y = m mx + b slope = 7 + 7

1 1 Opposite Reciprocal 2010 Preston PowerPoints Whiteboard Practice 15) Write an equation in slope-intercept form of a line that has a y-intercept of 3 & that is perpendicular to the line y = 7x 2. slope (m) = 7 y = m mx + b 1 slope = 7 +7 1 1 y= x + 3b 7 2010 Preston PowerPoints

Whiteboard Practice 15) Write an equation in slope-intercept form of a line that has a y-intercept of 3 & that is perpendicular to the line y = 7x 2. slope (m) = 7 y = m mx + b 1 slope = 7 +7 1 1 y= x+3 7 2010 Preston PowerPoints Example 16) Write an equation in slope-intercept form of a line that passes through the point

49 (4, 9) & that is perpendicular 2 + 3. 3 to the line y = 2x slope (m) = 2 y=m mx + b slope = 2 2 1 1 Opposite Reciprocal 2010 Preston PowerPoints Example 16) Write an equation in slope-intercept form of a line that passes through the point 49 (4, 9) & that is perpendicular to the line y = 2x + 3.

slope (m) = 2 y=m mx + b 1 slope = 2 2 1 1 y= x+ b 2 2010 Preston PowerPoints Example 16) Write an equation in slope-intercept form of a line that passes through the point 49 (4, 9) & that is perpendicular to the line y = 2x + 3. 1

y=m mx + b y= x + b7 2 9 = 2 + b + 2 + 2 1 9 = (4) + b 2 7 = b 2010 Preston PowerPoints Example 16) Write an equation in slope-intercept form of a line that passes through the point (4, 9) & that is perpendicular

to the line y = 2x + 3. 1 y=m mx + b y= x 7 2 9 = 2 + b + 2 + 2 1 9 = (4) + b 2 7 = b 2010 Preston PowerPoints Whiteboard Practice 17) Write an equation in slope-intercept form of a line that

passes through the point 13 (1, 3) & that is parallel to the 1 line y = 1 . 1x + 6 slope (m) = 1 y = m mx + b slope = 1 Same slope 1 y= x + b 2010 Preston PowerPoints Whiteboard Practice 17) Write an equation in slope-intercept form of a line that passes through the point 13 (1, 3) & that is parallel to the

1 line y = 1x + . 6 y=m mx + b y = x + b2 3 = 1 + b 3 = (1) + b + 1 +1 2 = b

2010 Preston PowerPoints Whiteboard Practice 17) Write an equation in slope-intercept form of a line that passes through the point (1, 3) & that is parallel to the 1 line y = 1x + . 6 y=m mx + b y= x2 3 = 1 + b 3 = (1) + b + 1

+1 2 = b 2010 Preston PowerPoints Example 18) Write an equation in slope-intercept form of a line that passes through the point 3 8 & that is parallel to the (3,8) 11 line y = x 3. 3 y=m mx + b 4 slope (m) =

slope = y= 1 4 x+ b 11 4 1 Same slope 4 2010 Preston PowerPoints Example 18) Write an equation in slope-intercept form of a line that passes through the point 3 8 & that is parallel to the (3,8) 1 line y = x 3.

y= 8 = 4 1 4 x + b4 1 4 (3) + b y=m mx + b 8 = 12 + b 12 12 4 = b

2010 Preston PowerPoints Example 18) Write an equation in slope-intercept form of a line that passes through the point (3,8) & that is parallel to the 1 line y = x 3. y= 8 = 4 1 4 x4 1 4 (3) + b

y=m mx + b 8 = 12 + b 12 12 4 = b 2010 Preston PowerPoints Whiteboard Practice 19) Write an equation in slope-intercept form of a line that 6 5) 5 & passes through the point (6, that is perpendicular to the 33 line y = x 2. 2 m

y = mx + b 7 Opposite 33 slope (m) = Reciprocal 7 slope = 3 7 3 7 2010 Preston PowerPoints Whiteboard Practice 19) Write an equation in slope-intercept form of a line that 6 5)

5 & passes through the point (6, that is perpendicular to the 3 line y = x 2. y = m mx + b 7 Opposite 3 slope (m) = Reciprocal 7 slope = 3 7 7 y= x + b 3 7 3

2010 Preston PowerPoints Whiteboard Practice 19) Write an equation in slope-intercept form of a line that 6 5) 5 & passes through the point (6, that is perpendicular to the 3 line y = x 2. y = m mx + b y= 7 3 5 = 7 3

7 x + 9b (6) + b 5 = 14 + b + 14 + 14 9 = b 2010 Preston PowerPoints Whiteboard Practice 19) Write an equation in slope-intercept form of a line that passes through the point (6, 5) & that is perpendicular to the 3 line y =

x 2. y = m mx + b y= 7 3 5 = 7 3 7 x+9 (6) + b 5 = 14 + b + 14 + 14 9 =

b 2010 Preston PowerPoints Example 20) Write an equation in slope-intercept form of a line that passes through the point 2 7 & that is parallel to the (2,7) line y = 3x 4 3 + 4. slope (m) = 3 y=m mx + b slope = 3 Same slope 1 y= 3x + b 2010 Preston PowerPoints

Example 20) Write an equation in slope-intercept form of a line that passes through the point 2 7 & that is parallel to the (2,7) line y = 3x + 4. y=m mx + b y = 3x + 1b 7 = 6 + b 7 = 3 (2) + b 6 6 1 = b 2010 Preston PowerPoints

Example 20) Write an equation in slope-intercept form of a line that passes through the point (2,7) & that is parallel to the line y = 3x + 4. y=m mx + b y = 3x + 1 7 = 6 + b 7 = 3 (2) + b 6 6 1 = b 2010 Preston PowerPoints

Whiteboard Practice 21) Write an equation in slope-intercept form of a line that passes through the point 10 4) 4 & that is perpendicular (10, 5 + 7. 7 to the line y = 5x slope (m) = 5 y = m mx + b 5 slope = 5 1 1 Opposite Reciprocal 2010 Preston PowerPoints Whiteboard Practice 21) Write an equation in slope-intercept form of a line

that passes through the point 10 4) 4 & that is perpendicular (10, to the line y = 5x + 7. slope (m) = 5 y = m mx + b 1 slope = - 5 5 1 1 y= x+ b 5 2010 Preston PowerPoints Whiteboard Practice 21) Write an equation in slope-intercept form of a line that passes through the point 10 4) 4 & that is perpendicular

(10, to the line y = 5x + 7. 1 y=m mx + b y= x + b6 5 4 = 2+ b 2 2 1 4 = (10) + b 5 6 = b 2010 Preston PowerPoints Whiteboard Practice 21) Write an equation in slope-intercept form of a line that passes through the point (10, 4) & that is perpendicular to the line y = 5x + 7.

1 y=m mx + b y= x 6 5 4 = 2+ b 2 2 1 4 = (10) + b 5 6 = b 2010 Preston PowerPoints Whiteboard Practice 22) Write an equation in slope-intercept form of a line that passes through the point 4,2) 4 2 & that is parallel to the ( 11

7 line y = x .y=m mx + b 3 slope (m) = 11 3 8 slope = 1 Same slope 3 1 y= x + b 3 2010 Preston PowerPoints Whiteboard Practice

22) Write an equation in slope-intercept form of a line that passes through the point 4,2) 4 2 & that is parallel to the ( 1 7 line y = x .y=m mx + b y= 1 3 2 = 1 3 3

2 x + 3b (4)+ b 8 6 2 = 3 4 3 2 3 4 3 4

3 = + b b 2010 Preston PowerPoints Whiteboard Practice 22) Write an equation in slope-intercept form of a line that passes through the point ( 4,2) & that is parallel to the 1 7 line y = x .y=m mx + b y=

1 3 2 = 1 3 3 x+ 2 3 (4)+ b 8 6 2 =

3 4 3 2 3 4 3 4 3 = + b b 2010 Preston PowerPoints Example

23) Write an equation in slope-intercept form of a line that 34 & passes through the point (3,4) that is perpendicular to the 22 44 line y = x .y=m mx + b 3 5 Opposite 22 slope (m) = Reciprocal 3 slope = 2 3

2 3 2010 Preston PowerPoints Example 23) Write an equation in slope-intercept form of a line that 34 & passes through the point (3,4) that is perpendicular to the 2 4 line y = x .y=m mx + b 3 5 Opposite 2 slope (m) =

Reciprocal 3 y= slope = 2 3 3 x+ b 2 3 2 2010 Preston PowerPoints Example 23) Write an equation in slope-intercept form of a line that 34 & passes through the point (3,4) that is perpendicular to the 2 4

line y = x .y=m mx + b y= 4 = 3 2 3 x+ 3 2 1 b 2 (3) + b 5

8 2 4 = 9 2 1 2 9 2 9 2 = + b

b 2010 Preston PowerPoints Example 23) Write an equation in slope-intercept form of a line that passes through the point (3,4) & that is perpendicular to the 2 4 line y = x .y=m mx + b y= 4 = 3 2 3

x 3 2 1 2 (3) + b 5 8 2 4 = 9 2 1 2 9

2 9 2 = + b b 2010 Preston PowerPoints Example 24) Write an equation in slope-intercept form of a line that has a y-intercept of 3 & that is perpendicular to the y=m mx + b 10 line 3x + 2y = 10. 3x + 2y =

10 3x 3x 2y = 3x + 10 2 2 2 3 y= x+5 2 2010 Preston PowerPoints Example 24) Write an equation in slope-intercept form of a line that has a y-intercept of 3 & that is perpendicular to the line 3x + 2y = 10. 3 y=m mx + b slope (m) =

2 3 3 slope = 2 2 Opposite Reciprocal 2010 Preston PowerPoints Example 24) Write an equation in slope-intercept form of a line that has a y-intercept of 3 & that is perpendicular to the line 3x + 2y = 10. 3 y=m mx + b slope (m) = 2 2 3

slope = 2 2 3 y= x + b3 3 2010 Preston PowerPoints Example 24) Write an equation in slope-intercept form of a line that has a y-intercept of 3 & that is perpendicular to the line 3x + 2y = 10. 3 y=m mx + b slope (m) = 2 2 3 slope = 2 2 3 y=

x3 3 2010 Preston PowerPoints Whiteboard Practice 25) Write an equation in slope-intercept form of a line that has a y-intercept of 8 & that is perpendicular to the line m +b y = mx 8x 8x++2y 2y==12 12. 8x + 2y = 12 + 8x + 8x 2y = 8x + 12 2 2

2 y = 4x + 6 2010 Preston PowerPoints Whiteboard Practice 25) Write an equation in slope-intercept form of a line that has a y-intercept of 8 & that is perpendicular to the line 4 slope 8x + 2y = 12. y=m mx + b (m) = 1 4 4 slope = 1

1 Opposite Reciprocal 2010 Preston PowerPoints Whiteboard Practice 25) Write an equation in slope-intercept form of a line that has a y-intercept of 8 & that is perpendicular to the line 8x + 2y = 12. 4 y=m mx + b slope (m) = 1 4 1 slope = 11

4 y = x + 8b 4 2010 Preston PowerPoints Whiteboard Practice 25) Write an equation in slope-intercept form of a line that has a y-intercept of 8 & that is perpendicular to the line 8x + 2y = 12. 4 y=m mx + b slope (m) = 1 4 1 slope = 11 4 y= x+8

4 2010 Preston PowerPoints Example 26) Write an equation in point-slope form for the line 5 4) 4 & that contains the point (5, that is perpendicular to the 15 line 18x + 3y = 15. 18x + 3y = 15 + 18x + 18x 3y = 18x + 15 3 3 3 y = 6x + 5 2010 Preston PowerPoints

Example 26) Write an equation in point-slope form for the line 5 4) 4 & that contains the point (5, that is perpendicular to the line 18x + 3y = 15. 6 slope (m) = 1 6 6 slope = 1 1 Opposite Reciprocal 2010 Preston PowerPoints

Example 26) Write an equation in point-slope form for the line 5 4) 4 & that contains the point (5, that is perpendicular to the line 18x + 3y = 15. 6 slope (m) = 1 6 1 slope = 1 1 6 y y14 = (x x51 ) 6 2010 Preston PowerPoints Example 26) Write an equation in

point-slope form for the line that contains the point (5, 4) & that is perpendicular to the line 18x + 3y = 15. 6 slope (m) = 1 6 1 slope = 11 6 y + 4 = (x 5) 6 2010 Preston PowerPoints Whiteboard Practice 27) Write an equation in point-slope form for the line 2,6) 26 & that contains the point (

that is perpendicular to the line y = 3 x + 5 4 3 y= x+5 4 2010 Preston PowerPoints Whiteboard Practice 27) Write an equation in point-slope form for the line 2,6) 26 & that contains the point ( that is perpendicular to the line 3 slope (m) = 4 3 3 slope = 4

4 Opposite Reciprocal 2010 Preston PowerPoints Whiteboard Practice 27) Write an equation in point-slope form for the line 2,6) 26 & that contains the point ( that is perpendicular to the line 3 slope (m) = 4 3 4 slope = 44 3

y y61 = (x x12) 3 2010 Preston PowerPoints Whiteboard Practice 27) Write an equation in point-slope form for the line that contains the point ( 2,6) & that is perpendicular to the line 3 slope (m) = 4 3 4 slope = 44 3 y6= (x + 2) 3 2010 Preston PowerPoints

Example 28) Write an equation in point-slope form for the line 52 & that contains the point (5,2) that is perpendicular to the 6 line 3y = 6. 3y = 6 3 3 y=2 2010 Preston PowerPoints Example 28) y = 2 Special line ( 3 ,2) ( 2 ,2) -5 (x,y) y

Rise 0 = = 0 5 Run 5 4 0 ( 3,2) (2,2) 3 2 +5 1 -4 -3 -2 -1

-1 1 2 3 4 5 -2 -3 -4 -5 2010 Preston PowerPoints Example 28) Write an equation in point-slope form for the line that contains the point (5,2) 52 & that is perpendicular to the

line 3y = 6. 0 slope (m) = 1 0 0 slope = 1 1 2010 Preston PowerPoints Example 28) Write an equation in point-slope form for the line that contains the point (5,2) 52 & that is perpendicular to the line 3y = 6. 0 slope (m) = 1 0 1

slope = 1 0 This lines slope is u defined 2010 Preston PowerPoints Review/Example 1 = u defined 0 You take the x value 5 Special line 4 (3, 2) (3, 1) (x,y) x x = 3 3

(3,2) 2 1 -5 -4 -3 -2 -1 -1 -2 1 2 3 4 5 (3, 1) -3

-4 -5 2010 Preston PowerPoints Example 28) Write an equation in point-slope form for the line 52 & that contains the point (5,2) that is perpendicular to the line 3y = 6. Where are we going to get an x value on the line? x= x 5 value 2010 Preston PowerPoints Example 28) Write an equation in point-slope form for the line that contains the point (5,2) & that is perpendicular to the

line 3y = 6. Where are we going to get an x value on the line? x= 5 2010 Preston PowerPoints Whiteboard Practice 29) Graph the line that 3,4) 3 4 & is contains the point ( perpendicular to the line 5 2 4 y= x+2 2. 3 3 2 2 slope (m) =

1 3 -5 -4 -3 -2 -1 1 2 3 4 -1 2 slope = -2 -3 2 3 -4 -5 3 5 2010 Preston PowerPoints Whiteboard Practice 29) Graph the line that 3,4) 3 4 & is contains the point (

perpendicular to the line 5 2 ( 3,4) 4 y= x + 2. 3 3 -3 2 2 +2 slope (m) = 1 3 -5 -4 -3 -2 -1 1 2 3 4 -1 2 slope = -2 -3 Rise 3 3 = -4

-5 2 Run 5 2010 Preston PowerPoints Key Points & Dont Forget When finding a perpendicular slope, dont forget to do the opposite sign. ( 2 becomes positive) 2010 Preston PowerPoints The Assignment pg. 261-263 #s 17-65 odd, 70-84 even 2010 Preston PowerPoints Bibliography Please email [email protected] with errors,

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