Transcript Parallel and Perpendicular Lines

Parallel and Perpendicular Lines
Section 5-6
Goals
Goal
• To determine whether lines
are parallel, perpendicular,
or neither.
• To write linear equations of
parallel lines and
perpendicular lines.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Parallel lines
• Perpendicular lines
• Opposite reciprocals
Definition
• Parallel Lines - are lines in the same plane that
never intersect.
Line WX is parallel to line YZ.
WX || YZ.
Parallel Lines
These two lines are parallel.
Parallel lines are lines in the
same plane that have no
points in common. In other
words, they do not intersect.
As seen on the graph, parallel lines have the same slope.
For y = 3x + 100, m = 3 and for y = 3x + 50, m = 3.
Parallel Lines
Example: Identifying
Parallel Lines
Identify which lines are parallel.
The lines described by
and
both have slope
These lines are parallel. The lines
described by y = x and y = x + 1
both have slope 1. These lines are
parallel.
.
Example: Identifying
Parallel Lines
Identify which lines are parallel.
Write all equations in slope-intercept form to determine the slope.
y = 2x – 3 slope-intercept form
slope-intercept form


Example: Continued
Identify which lines are parallel.
Write all equations in slope-intercept form to determine the slope.
2x + 3y = 8
–2x
– 2x
3y = –2x + 8
y + 1 = 3(x – 3)
y + 1 = 3x – 9
–1
–1
y = 3x – 10
Example: Continued
The lines described by y = 2x – 3
and y + 1 = 3(x – 3) are not parallel with
any of the lines.
y = 2x – 3
The lines described by
and
represent parallel lines. They
each have the slope
.
y + 1 = 3(x – 3)
Your Turn:
Identify which lines are parallel.
y = 2x + 2; y = 2x + 1; y = –4; x = 1
The lines described by
y = 2x + 2 and y = 2x + 1
represent parallel lines. They
each have slope 2.
y = 2x + 2
y = 2x + 1
Equations x = 1 and y = –4
are not parallel.
y = –4
x=1
Your Turn:
Identify which lines are parallel.
Write all equations in slope-intercept form to determine the slope.
y = 3x
Slope-intercept form

Slope-intercept form

Your Turn: Continued
Identify which lines are parallel.
Write all equations in slope-intercept form to determine the slope.
–3x + 4y = 32
+3x
+3x
4y = 3x + 32
y – 1 = 3(x + 2)
y – 1 = 3x + 6
+1
+1
y = 3x + 7
Your Turn: Continued
The lines described by
–3x + 4y = 32 and y =
–3x + 4y = 32
+ 8 have
the same slope, but they are not
y = 3x
parallel lines. They are the same line.
The lines described by
y = 3x and y – 1 = 3(x + 2) represent
parallel lines. They each have slope 3.
y – 1 = 3(x + 2)
Definition
• Perpendicular Lines – lines that intersect to form 90o
angles, or right angles.
Line RS is perpendicular to line TU.
RS __| TU.
Perpendicular Lines
Perpendicular lines have slopes that are opposite reciprocals.
Definition
• Opposite Reciprocals – two numbers whose
product is – 1.
– Example:
• To find the opposite reciprocal (also called negative
reciprocal) of – 3/4, first find the reciprocal, –4/3.
• Then write its opposite, 4/3.
• Since -3/4 ∙ 4/3 = -1, therefore 4/3 is the opposite
reciprocal of -3/4.
Opposite Reciprocals
Helpful Hint
If you know the slope of a line, the slope of a
perpendicular line will be the "opposite
reciprocal.”
Example: Indentifying
Perpendicular Lines
Identify which lines are perpendicular: y = 3; x = –2; y = 3x;
.
The graph described by y = 3 is a
horizontal line, and the graph
described by x = –2 is a vertical line.
These lines are perpendicular.
The slope of the line described by y =
3x is 3. The slope of the line described
by
is
.
x = –2
y=3
y =3x
Example: Continued
Identify which lines are perpendicular: y = 3; x = –2; y = 3x;
.
x = –2
y=3
These lines are perpendicular
because the product of their
slopes is –1.
y =3x
Your Turn:
Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4);
x = 3; y =
The graph described by x = 3 is a
vertical line, and the graph described
by y = –4 is a horizontal line. These
lines are perpendicular.
The slope of the line described by
y – 6 = 5(x + 4) is 5. The slope of the
line described by
y=
is
x=3
y = –4
y – 6 = 5(x + 4)
Your Turn: Continued
Identify which lines are perpendicular: y = –4; y – 6 = 5(x + 4);
x = 3; y =
x=3
These lines are perpendicular
because the product of their
slopes is –1.
y = –4
y – 6 = 5(x + 4)
Example: Writing Equations of
Parallel and Perpendicular Lines
Write an equation in slope-intercept form for the line that passes
through (4, 10) and is parallel to the line described by y = 3x + 8.
Step 1 Find the slope of the line.
The slope is 3.
y = 3x + 8
The parallel line also has a slope of 3.
Step 2 Write the equation in point-slope form.
y – y1 = m(x – x1)
Use the point-slope form.
y – 10 = 3(x – 4)
Substitute 3 for m, 4 for x1,
and 10 for y1.
Example: Continued
Step 3 Write the equation in slope-intercept form.
y – 10 = 3(x – 4)
y – 10 = 3x – 12)
y = 3x – 2
Distribute 3 on the right side.
Add 10 to both sides.
Example: Writing Equations of
Parallel and Perpendicular Lines
Write an equation in slope-intercept form for the line that passes
through (2, –1) and is perpendicular to the line described by
y = 2x – 5.
Step 1 Find the slope of the line.
The slope is 2.
y = 2x – 5
The perpendicular line has a slope of
because
Step 2 Write the equation in point-slope form.
y – y1 = m(x – x1)
Use the point-slope form.
Substitute
2 for x1.
for m, –1 for y1, and
Example: Continued
Step 3 Write the equation in slope-intercept form.
Distribute
on the right side.
Subtract 1 from both sides.
Your Turn:
Write an equation in slope-intercept form for the line that passes
through (5, 7) and is parallel to the line described by y = x – 6.
Step 1 Find the slope of the line.
y=
x –6
The parallel line also has a slope of
The slope is
.
.
Step 2 Write the equation in point-slope form.
y – y1 = m(x – x1)
Use the point-slope form.
Your Turn: Continued
Step 3 Write the equation in slope-intercept form.
Distribute
on the right side.
Add 7 to both sides.
Your Turn:
Write an equation in slope-intercept form for the line that passes
through (–5, 3) and is perpendicular to the line described by
y = 5x.
Step 1 Find the slope of the line.
The slope is 5.
y = 5x
The perpendicular line has a slope of
because
.
Step 2 Write the equation in point-slope form.
Use the point-slope form.
y – y1 = m(x – x1)
Your Turn: Continued
Write an equation in slope-intercept form for the line that passes
through (–5, 3) and is perpendicular to the line described by
y = 5x.
Step 3 Write in slope-intercept form.
Distribute
on the right side.
Add 3 to both sides.
Joke Time
• What do clouds wear under their shorts?
• Thunderpants!
• What kind of music do mummies listen to?
• Wrap!
• Why is there no gambling in Africa?
• Too many Cheetahs!
Assignment
• 5-6 Exercises Pg. 361 - 362: #6 – 40 even