introduction the slopes of parallel lines are always equal, whereas the slopes of perpendicular...
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IntroductionThe slopes of parallel lines are always equal, whereas the slopes of perpendicular lines are always opposite reciprocals. It is important to be able to determine whether lines are parallel or perpendicular, but the creation of parallel and perpendicular lines is also important. In this lesson, you will write the equations of lines that are parallel and perpendicular to a given line through a given point.
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6.1.3: Working with Parallel and Perpendicular Lines
Key Concepts• You can write the equation of a line through a given
point that is parallel to a given line if you know the equation of the given line. It is necessary to identify the slope of the given equation before trying to write the equation of the line that is parallel or perpendicular.
•Writing the given equation in slope-intercept form allows you to quickly identify the slope, m, of the equation.
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6.1.3: Working with Parallel and Perpendicular Lines
Key Concepts, continued• If the given equation is not in slope-intercept form,
take a few moments to rewrite it.
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Writing Equations Parallel to a Given Line Through a Given Point
1. Rewrite the given equation in slope-intercept form if necessary. 2. Identify the slope of the given line. 3. Write the general point-slope form of a linear equation:
y – y1 = m(x – x1).4. Substitute the slope of the given line for m in the general
equation.5. Substitute x and y from the given point into the general
equation for x1 and y1. 6. Simplify the equation.7. Rewrite the equation in slope-intercept form if necessary.
6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice
Example 1Write the slope-intercept form of an equation for the line that passes through the point (5, –2) and is parallel to the graph of 8x – 2y = 6.
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6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 1, continued
1. Rewrite the given equation in slope-intercept form.
8x – 2y = 6 Given equation
–2y = 6 – 8x Subtract 8x from both sides.
y = –3 + 4x Divide both sides by –2.
y = 4x – 3 Write the equation in slope-intercept form.
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6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 1, continued
2. Identify the slope of the given line.The slope of the line y = 4x – 3 is 4.
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6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 1, continued
3. Substitute the slope of the given line for m in the point-slope form of a linear equation.
y – y1 = m(x – x1) Point-slope form
y – y1 = 4(x – x1) Substitute m from the given
equation.
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6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 1, continued
4. Substitute x and y from the given point into the equation for x1 and y1.
y – y1 = 4(x – x1) Equation
y – (–2) = 4(x – 5) Substitute (5, –2) for x1 and y1.
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6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 1, continued
5. Simplify the equation.y – (–2) = 4(x – 5) Equation with substituted
values for x1 and y1
y – (–2) = 4x – 20 Distribute 4 over (x – 5).
y + 2 = 4x – 20 Simplify.
y = 4x – 22 Subtract 2 from both sides.
The equation of the line through the point (5, –2) that is parallel to the equation 8x – 2y = 6 is y = 4x – 22. 9
6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 1, continuedThis can be seen on the following graph.
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✔6.1.3: Working with Parallel and Perpendicular Lines
Key Concepts, continued• Writing the equation of a line perpendicular to a given
line through a given point is similar to writing equations of parallel lines.
• The slopes of perpendicular lines are opposite reciprocals.
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6.1.3: Working with Parallel and Perpendicular Lines
Key Concepts, continued
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Writing Equations Perpendicular to a Given Line Through a Given Point
1. Rewrite the given equation in slope-intercept form if necessary. 2. Identify the slope of the given line. 3. Find the opposite reciprocal of the slope of the given line. 4. Write the general point-slope form of a linear equation:
y – y1 = m(x – x1).5. Substitute the opposite reciprocal of the given line for m in the
general equation.6. Substitute x and y from the given point into the general
equation for x1 and y1. 7. Simplify the equation.8. Rewrite the equation in slope-intercept form if necessary.
6.1.3: Working with Parallel and Perpendicular Lines
Key Concepts, continued• The shortest distance between two points is a line.
• The shortest distance between a given point and a given line is the line segment that is perpendicular to the given line through the given point.
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6.1.3: Working with Parallel and Perpendicular Lines
Key Concepts, continued
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Finding the distance from a point to a line.
1. Follow the steps outlined previously to find the equation of the line that is perpendicular to the given line through the given point.
2. Find the intersection between the two lines by setting the given equation and the equation of the perpendicular line equal to each other.
3. Solve for x. 4. Substitute the x-value into the equation of the given line to find
the y-value.
5. Find the distance between the given point and the point of
intersection of the given line and the perpendicular line using
the distance formula, .
6.1.3: Working with Parallel and Perpendicular Lines
Common Errors/Misconceptions• attempting to identify the slope of the given line
without transforming the equation into slope-intercept form
• incorrectly identifying the slope of the given line• incorrectly finding the slope of the line parallel to the
given line• incorrectly identifying the slope of the line
perpendicular to the given line• improperly substituting the x- and y-values into the
general point-slope equation
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6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice
Example 3
Find the point on the line 3x - 2y = 6 that is closest to the point (–2, 7).
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6.1.3: Working with Parallel and Perpendicular Lines
(Find the equation of the line perpendicular to the given line Passing through the point (–2, 7).)
Guided Practice: Example 3, continued
1. Find the line perpendicular to the given line, 3x - 2y = 6 , that passes through the point (–2, 7).
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6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 3, continued
2. Identify the slope of the given line. (rewrite in slope intercept form)
Slope = 18
6.1.3: Working with Parallel and Perpendicular Lines
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3y
63x2y -
6 2y -3x
x
2
3
Guided Practice: Example 3, continued
3. Find the opposite reciprocal of the slope of the given line.
The opposite of is .
The reciprocal of is .
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6.1.3: Working with Parallel and Perpendicular Lines
2
3
2
3
2
3
3
2
Guided Practice: Example 3, continued
4. Substitute the opposite reciprocal for m in the point-slope form of a linear equation.
y – y1 = m(x – x1) Point-slope form
Substitute m from the given equation.
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6.1.3: Working with Parallel and Perpendicular Lines
)(3
211 xxyy
Guided Practice: Example 3, continued
5. Substitute x and y from the given point into the equation for x1 and y1.
Equation
Substitute (–2, 7) for x1
and y1.
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6.1.3: Working with Parallel and Perpendicular Lines
)(3
211 xxyy
))2((3
27 xy
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Guided Practice: Example 3, continued
6. Simplify the equation.
Equation with substitutedvalues for x1 and y1
Distribute over (x – (–2)).
Add 7 to both sides.
6.1.3: Working with Parallel and Perpendicular Lines
))2((3
27 xy
3
4
3
27 xy
3
17
3
2 xy
3
2
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Guided Practice: Example 3, continued
The equation of the line through the point (–2, 7)
that is perpendicular to the graph of is
.
This can be seen on the following graph.
6.1.3: Working with Parallel and Perpendicular Lines
3- 3x -2y
3
17
3
2 xy
Guided Practice: Example 3, continued
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6.1.3: Working with Parallel and Perpendicular Lines
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Guided Practice: Example 3, continued
7. Find the intersection between the two lines by setting the given equation equal to the equation of the perpendicular line, then solve for x.
Set both equations equal to
each other.
Add 3 to both sides.
6.1.3: Working with Parallel and Perpendicular Lines
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3
3
17
3
2- xx
xx2
3
3
26
3
2-
Guided Practice: Example 3, continued
Add to both sides.
Divide both sides by .
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6.1.3: Working with Parallel and Perpendicular Lines
x3
2xx2
3
3
26
3
2-
x6
13
3
26
6
13
x4
Guided Practice: Example 3, continued
8. Substitute the value of x back into the given equation to find the value of y.
Given
equation
Substitute 4 for x.
y = 3 Simplify.
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6.1.3: Working with Parallel and Perpendicular Lines
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3y x
32
3y x
Guided Practice: Example 3, continued
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The point on the line
closest to (–2, 7) is
the point (4, 3)
.
6.1.3: Working with Parallel and Perpendicular Lines
Guided Practice: Example 3, continued
9. Calculate the distance between the two points using the distance formula.
Distance formula
Substitute values for
(x1, y1) and (x2, y2)
using (–2, 8) and (4, 3)
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6.1.3: Working with Parallel and Perpendicular Lines
22 )73()2(4(
Guided Practice: Example 3, continued
Simplify.
Evaluate squares.
Simplify. 30
6.1.3: Working with Parallel and Perpendicular Lines
22 )4()6(
1636
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Guided Practice: Example 3, continued
The distance between the point of intersection and
the given point is units, or approximately 7.2
units.
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6.1.3: Working with Parallel and Perpendicular Lines
✔
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