Parallel and Perpendicular Lines Online Tutorials:
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Transcript of Parallel and Perpendicular Lines Online Tutorials:
WELCOMETO
TOPIC :
www.freematheducation.com:
Parallel Lines
The online version of this tutorial can be accessed at www.freematheducation.com. It provides all the basics and the concept in details including more practice questions with complete step by step solutions.
In case you have any questions about algebra course outline, please feel free to contact us at [email protected] or call +91-94612-01100 (INDIA)
1
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Parallel Lines
Two lines with the same slope are said to be a parallel lines. When we plot a graph of parallel lines then they will never intersect to each other.
Line AB and Line CD are Parallel lines and it is denoted by AB||CD.
The equation of the straight lines are y = m1x + c and y = m2x + c are parallel if the slopes of both lines are equal i.e.
m1 = m2
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A B
C D
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Parallel Lines
Q. Are the lines 12x + 3y = – 9 and – 8x – 2y = 14 are parallel?
Solution:
First find the slope of both the lines
12x + 3y = – 9 –8x – 2y = 14
3y = –12x – 9 –2y = 8x + 14
y = –4x – 3 y = –4x – 7
Since the slopes of both the lines are equal, so the lines are parallel.
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Slope = –4 Slope = –4
y = –4x – 3
y = –4x – 7
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Parallel Lines
Practice Question :
Are the given lines are parallel?
1. 6x – 3y = 5 and 2y = 4x – 4
2. x + 2y = 6 and 2x – 2y = 6
3. 5x – y = 5 and y = 10x – 4
4. 3x – 3y = 6 and x – y = 6
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WELCOMETO
TOPIC :
www.freematheducation.com:
Constructing Parallel Lines
The online version of this tutorial can be accessed at www.freematheducation.com . It provides all the basics and the concept in details including more practice questions with complete step by step solutions.
In case you have any questions about algebra course outline, please feel free to contact us at [email protected] or call +91-94612-01100 (INDIA)
5
www.freematheducation.com:
Constructing Parallel Lines
We can find the equation of a line parallel to the a given line which is passing through the point by
finding the slope m of the given line;
Finding the equation of a line through the given point (x1, y1) with the slope equals to m.
The slope intercept form of a line is y = mx + c, where m is the slope of the line and c be its y intercept.
The equation of a line passing through point (x1, y1) with slope m is given by
y – y1 = m (x – x1)
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Constructing Parallel Lines
Q. Find the equation of a line which is parallel to the line 2x + 6y = 12 and passing through point (2, –1).
Solution:
To find the equation of a line, first find the slope of given line
6x + 2y = 12
2y = – 6x + 12
y = – 3x + 6
The slope of a line is -3. Since the slopes of parallel lines are equal. Therefore the equation of a line is given by
y – y1 = m (x – x1)
y – (–1) = –3 (x – 2)
y + 1= –3x + 6
3x + y = 5www.freematheducation.com 7
Slope = –3
Compare with
Slope Intercept Form
y = mx + c
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Constructing Parallel Lines
Practice Question :
1. Find the equation of the line going through the point (4, –1) and parallel to x + 2y = 10
2. Find the equation of the line going through the point (–2, –7) and parallel to 2x – 3y = 12
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WELCOMETO
TOPIC :
www.freematheducation.com:
Perpendicular Lines
The online version of this tutorial can be accessed at www.freematheducation.com . It provides all the basics and the concept in details including more practice questions with complete step by step solutions.
In case you have any questions about algebra course outline, please feel free to contact us at [email protected] or call +91-94612-01100 (INDIA)
9
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Perpendicular Lines
Perpendicular lines are the lines that intersect in a right angle.
We can find the equation of a line perpendicular to a given line and going through a given point by:
finding the slope m of the given line;
Finding the equation of a line through the given point (x1, y1) with the slope equals to -1/m.
The slope intercept form of a line is y = mx + c, where m is the slope of the line and c be its y intercept.
The equation of a line passing through point (x1, y1) with slope m is given by
y – y1 = m (x – x1)
The equation of the straight lines are y = m1x + c and y = m2x + c are perpendicular the slopes are negative reciprocal to each other i.e. m1m2 = -1.
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Perpendicular Lines
Q. Are the lines 2x + y = 5 and x – 2y = – 4 are perpendicular?
Solution:
First find the slope of both the lines
2x + y = 5 x – 2y = –4
y = –2x + 5 –2y = –x – 4
y = ½ x +2
Since the slopes of both the lines are negative reciprocal to each other, so both
the lines are perpendicular lines.
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Slope m1 = –2
Slope m2 = ½
2x + y = 5
x – 2y = – 4
m1 m2 = –2 x ½ = –1
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Perpendicular Lines
Practice Question :
Are the given lines are parallel?
1. 6x – 3y = 5 and 3x + 6y = 14
2. 3x – 3y = 6 and 3x + 3y = 8
3. x + 2y = 6 and x – 2y = 6
4. y = 10x – 12 and y = 10x – 4
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WELCOMETO
TOPIC :
www.freematheducation.com:
Constructing Perpendicular Lines
The online version of this tutorial can be accessed at www.freematheducation.com . It provides all the basics and the concept in details including more practice questions with complete step by step solutions.
In case you have any questions about algebra course outline, please feel free to contact us at [email protected] or call +91-94612-01100 (INDIA)
13
www.freematheducation.com:
Constructing Perpendicular Lines
We can find the equation of a line perpendicular to the given line which is passing through the point by
finding the slope m1 of the given line;
Find the negative reciprocal of that slope i.e. m2 = 1/m1
Finding the equation of a line through the given point (x1, y1) with the slope equals to m2.
The slope intercept form of a line is y = mx + c, where m is the slope of the line and c be its y intercept.
The equation of a line passing through point (x1, y1) with slope m2 is given by
y – y1 = m2 (x – x1)
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Constructing Perpendicular Lines
Q. Find the equation of a line given through the point (3, 0) and perpendicular to 6x + 2y = 12.
Solution:
To find the equation of a line, first find the slope of given line
6x + 2y = 12
2y = – 6x + 12
y = – 3x + 6
Slope of perpendicular line is m2 = –(1/ –3) = 1/3. Therefore the equation of a line perpendicular to given line is
y – y1 = m2 (x – x1)
y – (0) = 1/3 (x – 3)
y = –1/3 x – 1
x + 3y + 3 = 0
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Slope = –3
Compare with
Slope Intercept Form
y = mx + c
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Constructing Perpendicular Lines
Practice Question :
1. Find the equation of the line going through the point (4, –1) and perpendicular to x + 2y = 10.
2. Find the equation of the line going through the point (–2, –7) and perpendicular to 2x – 3y = 12
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WELCOMETO
TOPIC :
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Answer Key
The online version of this tutorial can be accessed at www.freematheducation.com . It provides all the basics and the concept in details including more practice questions with complete step by step solutions.
In case you have any questions about algebra course outline, please feel free to contact us at [email protected] or call +91-94612-01100 (INDIA)
17
www.freematheducation.com:
Answer Key
Your practice Session is completed. Your Answer Key is
Parallel Lines :
1. Yes 2. No 3. No 4. Yes
Constructing Parallel Lines :
1. x – y = 2 2. 2x – 3y = 17
Perpendicular Lines :
1. Yes 2. Yes 3. No 4. No
Constructing Perpendicular Lines :
1. 2x – y = 9 2. 3x + 2y = – 20
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Parallel Lines
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