Transcript Chapter 1

Chapter 11
Areas of Plane Figures
• Understand what is
meant by the area of a
polygon.
• Know and use the
formulas for the areas
of plane figures.
• Work geometric
probability problems.
11-1: Area of Rectangles
Objectives
• Learn and apply the area formula for a
square and a rectangle.
Math Notation for Different
Measurements
Dimensions
• Length (1 dimension)
Notation
• 1 unit - 2cm - 3in
– The length of a line is….
• Area (2 dimensions)
– The area of a rectangle is ….
• Volume (3 dimensions)
– The volume of a cube is….
• 2 units2
• 3 cm2– 10 in2
• 4 units3
• 8 cm3
Area
A measurement of the region covered by a geometric
figure and its interior.
What types of
jobs use area
everyday?
Postulate
The area of a square is the length of the side squared.
Area = s2
s
s
What’s the are of a square with..
• side length of 4?
• perimeter of 12 ?
Area Congruence Postulate
If two figures are congruent, then they have
the same area.
A
B
If triangle A is congruent to triangle B, then area A = area B.
With you partner: Why would congruent figures have the same area?
Area Addition Postulate
The area of a region is the sum of the areas of its
non-overlapping parts.
Area of figure =
Area A + Area B + Area C
B
A
C
Base (b)
• Any side of a rectangle or other parallelogram
can be considered to be a base.
Altitude (Height (h))
• Altitude to a base is any segment perpendicular to
the line containing the base from any point on the
opposite side.
• Called Height
Theorem
The area of a rectangle is the product of the base and
height.
h
Area = b x h
Using the variables shown on
the diagram create an equation
that would represent the
perimeter of the figure.
b
Remote Time
Classify each statement as True or False
Question 1
• If two figures have the same areas, then
they must be congruent.
Question 2
• If two figures have the same perimeter, then
they must have the same area.
Question 3
• If two figures are congruent, then they must
have the same area.
Question 4
• Every square is a rectangle.
Question 5
• Every rectangle is a square.
Question 6
• The base of a rectangle can be any side of
the rectangle.
White Board Practice
h
b
b
12m
h
3m
A
9cm
y-2
y
54 cm2
Group Practice
h
b
b
12m
9cm
y-2
h
3m
6cm
y
A
36m2
54 cm2
y2 – 2y
Find the area of the rectangle
3
5
AREA = 12
Group Practice
• Find the area of the figure. Consecutive
3
sides are perpendicular.
2
4
A = 114
2
units
5
6
5
11-2: Areas of Parallelograms,
Triangles, and Rhombuses
Objectives
• Determine and apply the area formula for a
parallelogram, triangle and rhombus.
Base (b) and Height (h)
PARTNERS….
• How do a rectangle and parallelogram relate?
• What could I do with this parallelogram to make it
look like a rectangle?
h
b
Theorem
The area of a parallelogram is the product of
the base times the height to that base.
**This right
triangle is key
to helping
solve!!
h
Area = b x h
b
Triangle Demo
• How can I take two congruent triangles and
connect them to make a new shape?
Theorem
The area of a triangle equals half the product
of the base times the height to that base.
A = bh
2
h
b
Partners
• How would you label the base and height of
these triangles?
Theorem
The area of a rhombus equals half the product
of the diagonals.
d1
d2
A = _________
d1∙d2
2
**WHAT DO YOU SEE
WITHIN THE DIAGRAM?
White Board Practice
• Find the area of the figure
6
3
A9 3
3
60º
6
White Board Practice
A  12
5
5
6
•
Just talk about this one
White Board Practice
• Find the area of the figure
A  30
12
5
13
•
Just talk about this one
White Board Practice
• Find the area of the figure
2
5
A  20
5
2
•
Just talk about this one
White Board Practice
• Find the area of the figure
A4 3
4
4
4
White Board Practice
• Find the area of the figure
5
4
5
5
4
5
A  24
Organization is Key
• Always draw the diagrams
• Know what parts of the formula you have
and what parts you need to find
• Right triangles will help you find missing
information
11-3: Areas of Trapezoids
Objectives
• Define and apply the area formula for a
trapezoid.
Trapezoid Review
A quadrilateral with exactly one pair of
parallel sides.
base
leg
median
base
leg
What type of trap
do we have if the
legs are
congruent?
Median
• Remember the median
is the segment that
connects the midpoints
of the legs of a
trapezoid.
• Length of median
= ½ (b1+b2)
b2
median
b1
The length of the median is the_______ of the bases.
Height
• The height of the trapezoid is the segment
that is perpendicular to the bases of the
trapezoid
b2
h
How do we measure
height for a trap?
Partners: Why is the
height perpendicular
to both bases?
b1
Labeling Height for Isosceles Trap
• Always label 2 heights when dealing with
an isosceles trap
Theorem
The area of a trapezoid equals half the product
of the height and the sum of the bases.
b2
h
b1
demo
White Board Practice
1. Find the area of the
trapezoid and the length
of the median
7
5
A = 50
Median = 10
13
White Board Practice
3. Find the area of the
trapezoid and the length
of the median
13
14
9
A = 138
Median = 11.5
12
Group Practice
• Find the area of the trapezoid
8
8
8
60º
Area = 48 3
Group Practice
• Find the area of the trapezoid
45º
3 2
4
Area =
33
2
Group Practice
• Find the area of the trapezoid
12
30º
30º
30
Area = 63 3
11.4 Areas of Regular Polygons
Objectives
• Determine the area of a regular polygon.
Regular Polygon Review
•All sides congruent
•All angles congruent
(n-2) 180
n
side
Circles and Regular Polygons
• Read Pg. 440 and 441
– Start at 2nd paragraph, “Given any circle…
• What does it mean that we can
inscribe a poly in a circle?
– Each vertex of the poly will be on the circle
Center of a regular polygon
is the center of the
circumscribed circle
center
Radius of a regular polygon
is the distance from
the center to a vertex
is the radius of the
circumscribed circle
Central angle of a regular polygon
Is an angle formed by two radii
drawn to consecutive vertices
Central angle
How many central angles does
this regular pentagon have?
How many central angles does a
regular octagon have?
Think – Pair – Share
Central angle
What connection do you
see between the 360◦
of a circle and the
measure of the central
angle of the regular
pentagon?
360
n
Apothem of a regular polygon
the perpendicular
distance from the center
to a side of the polygon
apothem
How many apothems does this
regular pentagon have?
How many apothems does a
regular triangle have?
Regular Polygon Review
**What do you think the
apothem does to the
central angle?
central angle
center
apothem
side
Perimeter = sum
of sides
Theorem
The area of a regular polygon is half the
product of the apothem and the perimeter.
What does each letter
represent in the diagram?
s = length of side
p = 8s
r
a
s
A = ap
2
RAPA
•
•
•
•
R adius
A pothem
P erimeter
A rea
r
a
s
This right triangle is the key to
finding each of these parts.
Radius, Apothem, Perimeter
1. Find the central angle 360
n
Radius, Apothem, Perimeter
2. Draw in the apothem… This divides the isosceles triangle
into two congruent right triangles
• How do we know it’s an isosceles triangle?
Radius, Apothem, Perimeter
r
a
x
3. Find the missing pieces
• What does ‘x’ represent?
Radius, Apothem, Perimeter
• Think 30-60-90
• Think 45-45-90
• Think SOHCAHTOA
r
A = ½ ap
a
p
A
8
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
r
A = ½ ap
a
8
p
4
A
24 3 48 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
IS THERE ANOTHER
AREA FORMULA FOR
THIS SHAPE?
r
a
p
A
A = ½ ap
5 2
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
x
a
r
A = ½ ap
5 2
r
x
a
a
p
5
40
A
100
IS THERE ANOTHER
AREA FORMULA FOR
THIS SHAPE?
r
A = ½ ap
a
p
A
8
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
a
x
r
A = ½ ap
a
8
p
A
4 3 48 96 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
a
x
r
a
p
A
A = ½ ap
24 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
a
x
r
A = ½ ap
4 3
a
p
6
A
24 3 72 3
1. Central angle
2. ½ of central angle
3. 45-45-90
30-60-90
SOHCAHTOA
r
a
x
11.5 Circumference and Areas of
Circles
Objectives
• Determine the circumference and area of a
circle.
C
   3.1415
d
r

• Greek Letter Pi (pronounced “pie”)
– Used in the 2 main circle formulas:
• Circumference and Area (What are these?)
• Pi is the ratio of the circumference of a circle to the
diameter.
• Ratio is constant for ALL CIRCLES
• Irrational number (cannot be expressed as a ratio of
two integers)
• Common approximations
– 3.14
– 22/7
Circumference
The distance around the outside of a circle.
**The Circumference
and the diameter have a
special relationship that
lead us to
=
C
d
Experiment
Circle Circumference Diameter
Number (nearest mm) (nearest mm)
1
2
3
4
5
Ratio of
Circumference/Diameter
(as a decimal)
Group Experiment
1. With the circular object
2. Using a piece of string measure around the
outside of one of the circles.
3. Using a ruler measure the piece of string
to the nearest mm.
4. Using a ruler measure the diameter to the
nearest mm.
5. Record in the table.
Experiment
6. Make a ratio of the Circumference.
Diameter
7. Give the ratio in decimal form to the
nearest hundredth.
8. Pass you circular object to the next group
and repeat
Experiment
Circle Circumference Diameter
Number (nearest mm) (nearest mm)
1
2
3
4
5
Ratio of
Circumference/Diameter
(as a decimal)
What do you think?
1. How does the measurement of the
circumference compare to the
measurement of the diameter?
2. Were there any differences in results? If
so, what were they?
3. Did you recognize a pattern? Were you
able to verify a pattern?
Circumference
The distance around the outside of a circle.
C=∏d
C = ∏ 2r
r
r
d
C = circumference
r = radius
d = diameter
Area
The area of a circle is the product of pi times
the square of the radius.
A r
2
For both formulas always
leave answers in

r
B
WHITEBOARDS
*put answers in terms of pi
r
d
C
A
15
8
26∏
100∏
18∏
Quiz review - Set up these
diagrams
1. A square with side 2√3
2. A rectangle with base √4 and diagonal √5
3. A parallelogram with sides 6 and 10 and a
45◦ angle
4. A rhombus with side 10 and a diagonal 12
5. An isosceles trapezoid with bases of 2 and
6 and base angles that measure 45 ◦
6. A regular hexagon with a perimeter 72
11.6 Arc Length and Areas of
Sectors
Objectives
• Solve problems about arc length and sector
and segment area.
A
r
B
Warm - up
1. If you had the two pizzas on
the right and you were really
hungry, which one would you
take a slice from? Why?
Same angle
Arc Measure
the measure of an arc is given by the measure
of its central angle.
A
C
AC
80
B
mAC  80
The central angle
tells us how much
of the 360 ◦ of crust
we are using from
our pizza.
The central angle
measure and arc
measure are the
same no matter the
size of circle.
Remember Circumference
The distance around the outside of a circle.
x◦
B
C
x◦
r
Finding the total
length
C   d  2 r
Arc Length
The length of the arc is part of the circle’s circumference…
the question is, what fraction of the total circumference
x◦
does it represent?
Circumference of circle
x◦
Degree measure of arc
LENGTH OF
ARC
x
(
)  2r
360
O
Example
If r = 5, what is the length of CB?
Measure of CB = 60◦
60 = 1
360
6
B
C
60◦
1 (2 ∙ 5) = 5 
3
6
O
Remember Area
A  r
2
B
C
Sector of a circle
aka – the area of the piece
of pizza
Area of a Sector
The area of a sector is part of the circle’s area… the
question is, what fraction of the total area does it
represent?
x◦
Area of circle
Degree measure of arc
x◦
O
AREA OF
SECTOR
x
2
(
)  r
360
Example
If r = 5, what is the area of sector
COB?
Measure of CB = 60◦
60 = 1
360
6
B
C
60◦
1 ( ∙ 52) = 25
6
6
O
REMEMBER!!!
• Both arc length and the area of the sector
are different with different size circles!
• Just think pizza
WHITEBOARDS
• ONE PARTNER OPEN BOOK TO PG. 453
(classroom exercises)
• ANSWER #2
– Length = 4
– Area = 12
• ANSWER # 4
– Length = 6
– Area = 12
• ANSWER #1 (we)
WHITEBOARDS
• Find the area of the shaded region
B
• 25∏ - 50
10
A
10
O
11-7 Ratios of Areas
Objectives
• Solve problems
about the ratios
of areas of
geometric
figures.
Ratio
• A ratio of one number to another is the quotient when the
first number is divided by the second.
• A comparison between numbers
• There are 3 different ways to express a ratio
1
2
3
5
1:2
3:5
a:b
1 to 2
3 to 5
a to b
a
b
Solving a Proportion
3 a

5 15
5a  45
a9
First, cross-multiply
Next, divide by 5
The Scale Factor
• If two polygons are similar, then they have a scale factor
• The reduced ratio between any pair of corresponding
sides or the perimeters.
• 12:3  scale factor of 4:1
12
**What have we used
scale factor for in past
chapters?
3
Comparing Areas of
Triangles
Two triangles with equal heights
4
4
Two triangles with equal heights
Partners: Compare the ratio of the areas to the ratio
of the bases
4
7
1
A  bh
2
4
3
Ratio of their areas
4
7
14
6
4
3
Ratio of areas = ?
4
7
7
3
4
3
Comparing Areas of Triangles
Rule # 1
If two triangles have equal
heights….
4
4
then the ratio of their areas
equals the ratio of their bases.
Two triangles with equal bases
Partners: Compare the
ratio of the areas to the
ratio of the heights
8
2
5
5
Ratio of Areas
8
20
5
2
5
5
Ratio of Areas = ?
8
4
1
2
5
5
Comparing Areas of Triangles
Rule # 2
If two triangles have equal
bases….
5
5
then the ratio of their areas
equals the ratio of their heights.
Comparing Areas of Triangles
Rule # 3
If two triangles are similar….
X
6
~
Y
With your partner:
What is the ratio of
the areas of X to Y?
4
then the ratio of their areas
equals the square (2) of the scale
factor.
Theorem
If the scale factor of two similar
figures is a:b, then…
1. the ratio of their perimeters is
a:b
2. the ratio of their areas is a2:b2.
~
7
Area = 27
3
Scale Factor- 7: 3
Ratio of P – 7: 3
Ratio of A – 49 :9
WHITEBOARDS
OPEN BOOK TO PG. 458
•
•
ANSWER #4
– Ratio of P – 1:3
– Ratio of A – 1:9
– If the smaller figure has an area of 3 what is the area of
the larger shape?
ANSWER # 10
– Scale factor – 4:7
–
•
(classroom exercises)
Ratio of P – 4:7
ANSWER # 13
a.
No b. ADE ~ ABC c. 4: 25 d. 4:21
WHITEBOARDS
• The areas of two similar triangles are 36
and 81. The perimeter of the smaller
triangle is 12. Find the perimeter of the
bigger triangle.
• 36/81 = 4/9  2/3 is the scale factor
• 2/3 = 12/x  x = 18
T or F
If two quadrilaterals are similar, then their
areas must be in the same ratio as the square
of the ratio of their perimeters
T
T or F
If the ratio of the areas of two equilateral
triangles is 1:3, then the ratio of the
perimeters is 1: 3
T
T or F
If the ratio of the perimeters of two rectangles
is 4:7, then the ratio of their areas must be
16:49
F
T or F
If the ratio of the areas of two squares is 3:2,
then the ratio of their sides must be 3 : 2
T
Remember
• Scale Factor a:b
• Ratio of perimeters a:b
• Ratio of areas a2:b2
11-8: Geometric
Probability
Solve problems
about geometric
probability
Read Pg. 461
• Solving Geometric Problems
using 2 principles
1. Probability of a point landing on a
certain part of a line (length)
2. Probability of a point landing in a
specific region of an area (area)
Sample Space
The number of
all possible
outcomes in a
random
experiment.
1. Total length of the line
2. Total area
Event:
A possible
outcome in a
random
experiment.
1. Specific segment of the line
2. Specific region of an area
Probability
The calculation of the
possible outcomes in
a random experiment
For example: When I pull a
popsicle stick from the
cup, what is the chance I
pull your name?
Event Space
P(e) 
Sample Space
Geometric Probability
1. The length of an event divided by the
length of the sample space.
•
In a 10 minute cycle a bus pulls up to a
hotel and waits for 2 minutes while
passengers get on and off. Then the bus
leaves. If a person walks out of the hotel
front door at a random time, what is the
probability that the bus is there?
Geometric Probability
2. The area of an event
divided by the area of
the sample space.
•
If a beginner shoots an
arrow and hits the
target, what is the
probability that the
arrow hits the red
bull’s eye?
1
2
3
WHITEBOARDS
OPEN BOOK TO PG. 462
•
ANSWER #2
–
•
1/3
ANSWER # 3
–
Give answer in terms of pi
(classroom exercises)
WHITEBOARDS
• Find the ratio of the areas
of WYV to XYZ
Y
– 4 to 49
• Find the ratio of the areas
of WYV to quad WVZX
2
– 4 to 45
W
• Find the probability of a
point from the interior of
XYZ will lie in the
interior of quad XWYZ
V
5
– 45/49
X
Z
Drawing Quiz- Set up these
diagrams
1. A rectangle with base 10 and diagonal 15
2. A parallelogram with sides 6 and 10 and a
60◦ angle
3. A rhombus with side 10 and a diagonal 12
4. An equilateral triangle with a perimeter = 27
5. Sector AOB: AO = 12 and the central angle
equals 50 degrees
6. Isosceles triangle with base of 10 and perimeter
of 40.
Test Review
Chapter Review
• 16
• 12
• 21
• 22
• Chapter test
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–
–
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4
9
12
15