geometry section 3-3 1112
DESCRIPTION
Slopes of LinesTRANSCRIPT
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Section 3-3Slopes of Lines
Monday, December 19, 2011
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Essential Questions
How do you find slopes of lines?
How do you use slope to identify parallel and perpendicular lines?
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Vocabulary
1. Slope:
2. Rate of Change:
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Vocabulary
1. Slope: The ratio of the vertical change to the horizontal change between two points
2. Rate of Change:
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Vocabulary
1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x
2. Rate of Change:
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Vocabulary
1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x
rise over run
2. Rate of Change:
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Vocabulary
1. Slope: The ratio of the vertical change to the horizontal change between two points; Change in y over change in x
rise over run
2. Rate of Change: A way to describe slope
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical5 Units
Distance between horizontal
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical5 Units
Distance between horizontal7 Units
Monday, December 19, 2011
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ExploreGraph the points A(−2, −4) and B(3, 3). Then draw both a vertical and horizontal line through both and determine the distance between the vertical lines, then between
the horizontal lines.y
x
A
B
Distance between vertical5 Units
Distance between horizontal7 Units
Up 7, Right 5
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Slope Formula
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Slope Formula
m =y2 − y1x2 − x1
for points
(x1,y1),(x2,y2 )
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Example 1Find the slope of the line that goes
through the following pairs of points.
a. C(−3, 4) and D(8, 1)
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Example 1Find the slope of the line that goes
through the following pairs of points.
a. C(−3, 4) and D(8, 1)(x1,y1)
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Example 1Find the slope of the line that goes
through the following pairs of points.
a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )
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Example 1Find the slope of the line that goes
through the following pairs of points.
a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )
m =y2 − y1x2 − x1
Monday, December 19, 2011
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Example 1Find the slope of the line that goes
through the following pairs of points.
a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )
m =y2 − y1x2 − x1
=1− 48 − (−3)
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Example 1Find the slope of the line that goes
through the following pairs of points.
a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )
m =y2 − y1x2 − x1
=1− 48 − (−3)
=−311
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Example 1Find the slope of the line that goes
through the following pairs of points.
a. C(−3, 4) and D(8, 1)(x1,y1) (x2,y2 )
m =y2 − y1x2 − x1
=1− 48 − (−3)
=−311
Down 3, Right 11
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Example 1Find the slope of the line that goes
through the following pairs of points.
b. E(5, −1) and F(−3, 7)
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Example 1Find the slope of the line that goes
through the following pairs of points.
b. E(5, −1) and F(−3, 7)
m =y2 − y1x2 − x1
Monday, December 19, 2011
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Example 1Find the slope of the line that goes
through the following pairs of points.
b. E(5, −1) and F(−3, 7)
m =y2 − y1x2 − x1
=7 − (−1)−3 − 5
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Example 1Find the slope of the line that goes
through the following pairs of points.
b. E(5, −1) and F(−3, 7)
m =y2 − y1x2 − x1
=7 − (−1)−3 − 5
=8−8
Monday, December 19, 2011
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Example 1Find the slope of the line that goes
through the following pairs of points.
b. E(5, −1) and F(−3, 7)
m =y2 − y1x2 − x1
=7 − (−1)−3 − 5
=8−8
= −1
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Example 1Find the slope of the line that goes
through the following pairs of points.
b. E(5, −1) and F(−3, 7)
m =y2 − y1x2 − x1
=7 − (−1)−3 − 5
=8−8
= −1
Down 1, Right 1
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Example 1Find the slope of the line that goes
through the following pairs of points.
c. G(−1, 2) and H(−1, 7)
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Example 1Find the slope of the line that goes
through the following pairs of points.
c. G(−1, 2) and H(−1, 7)
m =y2 − y1x2 − x1
Monday, December 19, 2011
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Example 1Find the slope of the line that goes
through the following pairs of points.
c. G(−1, 2) and H(−1, 7)
m =y2 − y1x2 − x1
=7 − 2
−1− (−1)
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Example 1Find the slope of the line that goes
through the following pairs of points.
c. G(−1, 2) and H(−1, 7)
m =y2 − y1x2 − x1
=7 − 2
−1− (−1)=50
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Example 1Find the slope of the line that goes
through the following pairs of points.
c. G(−1, 2) and H(−1, 7)
m =y2 − y1x2 − x1
=7 − 2
−1− (−1)=50
Undefined
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Example 1Find the slope of the line that goes
through the following pairs of points.
c. G(−1, 2) and H(−1, 7)
m =y2 − y1x2 − x1
=7 − 2
−1− (−1)=50
Up 5, Right 0
Undefined
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Example 1Find the slope of the line that goes
through the following pairs of points.
c. G(−1, 2) and H(−1, 7)
m =y2 − y1x2 − x1
=7 − 2
−1− (−1)=50
Up 5, Right 0
Undefined
Vertical LineMonday, December 19, 2011
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Example 1Find the slope of the line that goes
through the following pairs of points.
d. J(3, 4) and K(−2, 4)
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Example 1Find the slope of the line that goes
through the following pairs of points.
d. J(3, 4) and K(−2, 4)
m =y2 − y1x2 − x1
Monday, December 19, 2011
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Example 1Find the slope of the line that goes
through the following pairs of points.
d. J(3, 4) and K(−2, 4)
m =y2 − y1x2 − x1
=4 − 4−2 − 3
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Example 1Find the slope of the line that goes
through the following pairs of points.
d. J(3, 4) and K(−2, 4)
m =y2 − y1x2 − x1
=4 − 4−2 − 3
=0−5
Monday, December 19, 2011
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Example 1Find the slope of the line that goes
through the following pairs of points.
d. J(3, 4) and K(−2, 4)
m =y2 − y1x2 − x1
=4 − 4−2 − 3
=0−5
Up 0, Left 5
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Example 1Find the slope of the line that goes
through the following pairs of points.
d. J(3, 4) and K(−2, 4)
m =y2 − y1x2 − x1
=4 − 4−2 − 3
=0−5
Up 0, Left 5 Horizontal Line
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Example 1Find the slope of the line that goes
through the following pairs of points.
d. J(3, 4) and K(−2, 4)
m =y2 − y1x2 − x1
=4 − 4−2 − 3
=0−5
Up 0, Left 5 Horizontal Line
= 0
Monday, December 19, 2011
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Example 2In 2000, the annual sales for one
manufacturer of camping equipment was $48.9 million. In 2005, the total sales
were $85.9 million. If sales increase at the same rate, what will the total sales
be in 2015?
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Example 2In 2000, the annual sales for one
manufacturer of camping equipment was $48.9 million. In 2005, the total sales
were $85.9 million. If sales increase at the same rate, what will the total sales
be in 2015?
85.9 − 48.9 =
Monday, December 19, 2011
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Example 2In 2000, the annual sales for one
manufacturer of camping equipment was $48.9 million. In 2005, the total sales
were $85.9 million. If sales increase at the same rate, what will the total sales
be in 2015?
85.9 − 48.9 = 37
Monday, December 19, 2011
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Example 2In 2000, the annual sales for one
manufacturer of camping equipment was $48.9 million. In 2005, the total sales
were $85.9 million. If sales increase at the same rate, what will the total sales
be in 2015?
85.9 − 48.9 = 37Every 5 years, sales increase by $37 million
Monday, December 19, 2011
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Example 2In 2000, the annual sales for one
manufacturer of camping equipment was $48.9 million. In 2005, the total sales
were $85.9 million. If sales increase at the same rate, what will the total sales
be in 2015?
85.9 − 48.9 = 37Every 5 years, sales increase by $37 million
85.9 + 2(37) =
Monday, December 19, 2011
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Example 2In 2000, the annual sales for one
manufacturer of camping equipment was $48.9 million. In 2005, the total sales
were $85.9 million. If sales increase at the same rate, what will the total sales
be in 2015?
85.9 − 48.9 = 37Every 5 years, sales increase by $37 million
85.9 + 2(37) = 159.9
Monday, December 19, 2011
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Example 2In 2000, the annual sales for one
manufacturer of camping equipment was $48.9 million. In 2005, the total sales
were $85.9 million. If sales increase at the same rate, what will the total sales
be in 2015?
85.9 − 48.9 = 37Every 5 years, sales increase by $37 million
85.9 + 2(37) = 159.9
In 2015, sales should be about $159.9 million
Monday, December 19, 2011
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PostulatesSlopes of parallel lines:
Slopes of perpendicular lines:
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PostulatesSlopes of parallel lines:
Two lines will be parallel IFF they have the same slope. All vertical lines
are parallel.
Slopes of perpendicular lines:
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PostulatesSlopes of parallel lines:
Two lines will be parallel IFF they have the same slope. All vertical lines
are parallel.
Two lines will be perpendicular IFF the product of their slopes is −1. Vertical
and horizontal lines are perpendicular.
Slopes of perpendicular lines:
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
=−1− (−3)−2 −1
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
=−1− (−3)−2 −1
=2−3
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
=−1− (−3)−2 −1
=2−3
m(HJ ) = y2 − y1
x2 − x1
Monday, December 19, 2011
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
=−1− (−3)−2 −1
=2−3
m(HJ ) = y2 − y1
x2 − x1=3 − 06 − 5
Monday, December 19, 2011
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
=−1− (−3)−2 −1
=2−3
m(HJ ) = y2 − y1
x2 − x1=3 − 06 − 5
=31
Monday, December 19, 2011
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Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
=−1− (−3)−2 −1
=2−3
m(HJ ) = y2 − y1
x2 − x1=3 − 06 − 5
=31= 3
Monday, December 19, 2011
![Page 66: Geometry Section 3-3 1112](https://reader033.vdocuments.site/reader033/viewer/2022051611/54b2bcb14a7959ed058b46e2/html5/thumbnails/66.jpg)
Example 3Determine whether FG and HJ are
parallel, perpendicular, or neither for F(1, −3), G(−2, −1), H(5, 0), and J(6, 3).
Graph each line to verify your answer.
m(FG
) = y2 − y1x2 − x1
=−1− (−3)−2 −1
=2−3
m(HJ ) = y2 − y1
x2 − x1=3 − 06 − 5
=31= 3
Neither
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
m = −34
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
m = −34
Q
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
m = −34
Q
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
m = −34
Q
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
m = −34
Q
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
m = −34
Q
Monday, December 19, 2011
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y
x
Example 4Graph the line that contains Q(5, 1) and is parallel to the line through M(−2, 4)
and N(2, 1).
m =y2 − y1x2 − x1
=4 −1−2 − 2
=3−4
m = −34
QN
M
Monday, December 19, 2011
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Check Your Understanding
Use problems 1-11 to check the ideas from this lesson
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Problem Set
Monday, December 19, 2011
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Problem Set
p. 191 #13-39 odd
“I have found power in the mysteries of thought.” - Euripides
Monday, December 19, 2011