### Parallel and Perpendicular Lines

```Parallel and Perpendicular Lines
Perpendicular Lines
Parallel Lines
tutorial introduction
By Lindsay Hojnowski (2014)
Buffalo State College
04/2014
L. Hojnowski © 2014
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Aim for the Target
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.
04/2014
L. Hojnowski © 2014
2
Characteristics of Parallel Lines
Perpendicular Lines- Example 1
Quiz Question #1
Parallel Lines- Steps Given a
point and an equation
Perpendicular Lines- Example 2
Quiz Question #2
Parallel Lines- Example 1
Perpendicular Lines- Example 3
Quiz Question #3
Parallel Lines- Example 2
Determine whether parallel,
perpendicular, or neither- Steps
Quiz Question #4
Parallel Lines- Example 3
Determine- Example 1
Quiz Question #5
Characteristics of Perpendicular
Lines
Determine- Example 2
Perpendicular Lines- Steps
Given a point and an equation
Determine- Example 3
Quiz Question #6
Quiz Question #7
References
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L. Hojnowski © 2014
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Characteristics of Parallel Lines
Parallel lines:
1)
2)
Are lines that do not intersect
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
Parallel Lines- JMAP Video
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L. Hojnowski © 2014
4
Parallel Lines- Steps
Given a point and an equation
STEPS:
1) Rewrite the given equation into slopeintercept 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 – y1 = m (x – x1)
3) Distribute and simplify (if necessary)
4) Solve for y
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L. Hojnowski © 2014
Steps to writing a parallel line
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Parallel Lines- Example 1
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 – y1 = m (x – x1)
3) Distribute and simplify (if necessary)
4) Solve for y
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 = 4
2) y – y1 = m (x – x1)
y – 2 = 4 (x - - 2)
y – 2 = 4 (x + 2)
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3) y – 2 = 4 (x + 2)
y – 2 = 4x + 8
+2
+2
4) y = 4x + 10
L. Hojnowski © 2014
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Parallel Lines- Example 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 – y1 = m (x – x1)
3) Distribute and simplify (if necessary)
4) Solve for y
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 = 1/3
2) y – y1 = m (x – x1)
y – 4 = (1/3) (x - 6)
3) y – 4 = (1/3) (x - 6)
y – 4 = (1/3)x - 2
+4
+4
4) y = (1/3)x + 2
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L. Hojnowski © 2014
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Parallel Lines- Example 3
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 = ?
**In order to identify the slope, solve for y!
3x + y = 12
-3x
-3x
y = -3x +12
3) y – 6 = -3 (x – -1)
y – 6 = -3 (x + 1)
y – 6 = -3x - 3
m = -3
4) y – 6 = -3x - 3
+6
+6
y = -3x +3
2) y – y1 = m (x – x1)
y – 6 = -3 (x – -1)
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Example of a given point and a line
L. Hojnowski © 2014
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Characteristics of Perpendicular Lines
Perpendicular Lines:
1)
2)
Are lines that intersect at right angles
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
04/2014
L. Hojnowski © 2014
9
Perpendicular Lines- Steps
Given a point and an equation
STEPS:
1) Rewrite the given equation into slopeintercept 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 – y1 = m (x – x1)
3) Distribute and simplify (if necessary)
4) Solve for y
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L. Hojnowski © 2014
Steps to writing a perpendicular l line
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Perpendicular Lines- Example 1
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 – y1 = m (x – x1)
3) Distribute and simplify (if necessary)
4) Solve for y
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)
Perpendicular slope: -2
2) y – y1 = m (x – x1)
y – 2 = -2 (x - 4)
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L. Hojnowski © 2014
3) y – 2 = -2 (x - 4)
y – 2 = -2x + 8
+2
+2
4)
y= -2x + 10
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Perpendicular Lines- Example 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 perpendicular slope (negative reciprocal) in the
point-slope formula (y – y1 = m (x – x1)
3) Distribute and simplify (if necessary)
4) Solve for y
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)
Perpendicular slope: (-2/5)
2) y – y1 = m (x – x1)
y – -1 = (-2/5) (x - - 5)
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L. Hojnowski © 2014
3) y + 1= (-2/5) (x + 5)
y + 1= (-2/5)x - 2
-1
-1
4)
y= (-2/5)x - 3
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Perpendicular Lines- Example 3
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 – y1 = m (x – x1)
-2x
-2x
y – 6 = (3/2)(x – -4)
3y = -2x + 12
3
3
y = (-2/3)x + 4
m = -2/3
Perpendicular slope: (3/2)
3) y – 6 = (3/2)(x + 4)
y – 6 = (3/2)x + 6
+6
+6
4) y = (3/2)x + 12
Given a Point and a Line
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Determine whether parallel, perpendicular,
or neither- Steps
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
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Example of Perpendicular l LinesNegative Reciprocal Slope
L. Hojnowski © 2014
Example of Neither Parallel or Perpendicular Lines
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Determine whether parallel, perpendicular, or
neither- Example 1
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
Example1: Determine whether the graphs of the pair of equations are parallel, perpendicular,
or neither.
3x + 5y = 10
5x – 3y = -6
5x – 3y = -6
3x + 5y = 10
-5x
-5x
-3x
-3x
-3y = -5x - 6
5y = -3x + 10
-3
-3
5
5
y = (-5/-3)x + 2
y = (-3/5)x + 2
m = (5/3)
m = (-3/5)
PERPENDICULAR- they have
negative reciprocal slopes
04/2014
L. Hojnowski © 2014
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Determine whether parallel, perpendicular, or
neither- Example 2
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 2: Determine whether the graphs of the pair of equations are parallel, perpendicular,
or neither.
2x - 8y = -24
x – 4y = 4
x – 4y = 4
2x - 8y = -24
-x
-x
-2x
-2x
-4y = -x + 4
-8y = -2x - 24
-4
-4
-8
-8
y = (-1/-4)x - 1
y = (-2/-8)x + 3
m = (1/4)
m = (1/4)
PARALLEL- they have the same
slope
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L. Hojnowski © 2014
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Determine whether parallel, perpendicular, or
neither- Example 3
Example 3: Determine whether the graphs of the pair of equations are parallel, perpendicular,
or neither.
-3x + 4y = 8
-4x + 3y = -6
-4x + 3y = -6
+4x
+4x
3y = 4x - 6
3
3
y = (4/3)x - 2
m = (4/3)
-3x + 4y = 8
+3x
+3x
4y = 3x + 8
4
4
y = (3/4)x + 2
m = (3/4)
NEITHER- they aren’t the same slope
and are not negative reciprocals
They are reciprocals but not negative
reciprocals
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L. Hojnowski © 2014
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Quiz Question #1
1. What is the perpendicular slope of the line that passes through the
line:
y = (-3/4)x + 4 ?
a. -4/3
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b. 3/4
c. 4/3
L. Hojnowski © 2014
d. -3/4
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Quiz Question # 2
2. Which line is parallel to the line 4x + y = 3?
a. y = (1/4) x – 1
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b. y = 4x + 2
c. y = (-1/4) x – 6
L. Hojnowski © 2014
d. y = -4x + 5
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Quiz Question # 3
3. What is the slope of the line 2x + 7y = -35?
a. 2/7
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b. -2/7
c. 7/2
L. Hojnowski © 2014
d. -7/2
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Quiz Question # 4
4. Write an equation in slope-intercept form for the line that passes
through (0, 4) and is parallel to y = -4x + 5.
a. y = -4x
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b. y = -4x - 4
c. y = (1/4)x + 4
L. Hojnowski © 2014
d. y = -4x + 4
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Quiz Question # 5
5. 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. y = (-1/2)x - 4
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b. y = 2x + 16
c. y = -2x - 16
L. Hojnowski © 2014
d. y = (1/2)x + 4
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Quiz Question # 6
6. Determine whether 2x + 7y = -35 and 4x + 14y = -42 are parallel,
perpendicular, or neither.
a. neither
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b. parallel
L. Hojnowski © 2014
c. perpendicular
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Quiz Question # 7
6. Determine whether 3x + 5y = 10 and 5x – 3y= -6 are parallel,
perpendicular, or neither.
a. neither
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b. parallel
L. Hojnowski © 2014
c. perpendicular
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Reference from the dictionary
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 14th, 2012, from
• http://www.teambuildinggames.org/role-of-the-team-buildingfacilitator.
• Smiley Face, Retrieved on September 14th, 2012, from
http://ed101.bu.edu/StudentDoc/current/ED101fa10/rajensen/ima
ges/happy-face1.png.
• Wee, E. (2011). Try again, Retrieved on September 15th, 2012,