Transcript Section 3B Putting Numbers in Perspective

Section 10A
Fundamentals of Geometry
Pages 578-588
10-A
Perimeter and Area - Summary
10-A
Perimeter and Area
Rectangles
Perimeter
= l+ w+ l+ w
= 2l + 2w
Area
= length × width
=l×w
10-A
Perimeter and Area
Squares
Perimeter
= l+l+l+l
= 4l
Area
= length × width
=l×l
= l2
10-A
Perimeter and Area
Triangles
Perimeter
= a+b+c
Area
= ½×b×h
Perimeter and Area
Parallelograms
Perimeter
= l+ w+ l+ w
= 2l + 2w
Area
= length × height
= l×h
10-A
10-A
Perimeter and Area
Circles
Circumference(perimeter)
= 2πr
= πd
Area
= πr2
π ≈ 3.14159…
10-A
Practice with Area and Perimeter Formulas
Find the circumference/perimeter and area for each
figure described:
33/589 A circle with diameter 25 centimeters
Circumference = πd = π×25 cm= 25π cm
Area = πr2 = π×(25/2 cm)2 = 156.25π cm2
25
10-A
Practice with Area and Perimeter Formulas
Find the circumference/perimeter and area for each
figure described:
41/589 A rectangle with a length of 2 meters and a width
of 8 meters
Perimeter = 2m + 2m + 8m + 8m = 20 meters
Area = 2 meters × 8 meters = 16 meters2
8
2
10-A
Practice with Area and Perimeter Formulas
Find the circumference/perimeter and area for each
figure described:
37/589 A square with sides of length 12 miles
Perimeter = 12 km×4 = 48 miles
Area = (12 meters)2 = 144 miles2
12
12
10-A
Practice with Area and Perimeter Formulas
Find the circumference/perimeter and area for each figure
described:
39/589 A parallelogram with sides of length 10 ft and 20 ft
and a distance between the 20 ft sides of 5 ft.
Perimeter = 10ft +20 ft +10ft +20ft = 60ft
20
Area = 20ft × 5 ft = 100 ft2
10
5
10-A
Practice with Area and Perimeter Formulas
45/589 Find the perimeter and area of this triangle
9
Perimeter = 9+9+15 = 33 units
Area = ½ ×15×4 = 30 units2
4
15
9
Applications of
Area and Perimeter Formulas
47/589 A picture window has a length of 4 feet and a
height of 3 feet, with a semicircular cap on each end (see
Figure 10.20). How much metal trim is needed for the
perimeter of the entire window, and how much glass is
needed for the opening of the window?
49/589 Refer to Figure 10.14, showing the region to be
covered with plywood under a set of stairs. Suppose that
the stairs rise at a steeper angle and are 14 feet tall. What
is the area of the region to be covered in that case?
51/589 A parking lot is bounded on four sides by streets,
as shown in Figure 10.23. How much asphalt (in square
yards) is needed to pave the parking lot?
10-A
Surface Area and Volume
Practice with
Surface Area and Volume Formulas
10-A
79/591 Consider a softball with a radius of approximately 2 inches
and a bowling ball with a radius of approximately 6 inches.
Compute the surface area and volume for both balls.
Softball:
Surface Area = 4xπx(2)2 = 16π square inches
Volume = (4/3)xπx(2)3 = (32/3) π cubic inches
Bowling ball:
Surface Area = 4xπx(6)2 = 144π square inches
Volume = (4/3)xπx(6)3 = 288 π cubic inches
Practice with
Surface Area and Volume Formulas
ex6/585 Which holds more soup – a can with a diameter of 3
inches and height of 4 in, or a can with a diameter of 4 in and a
height of 3 inches?
Volume Can 1 = πr2h
= π×(1.5 in)2×4 in = 9π in3
Volume Can 2 = πr2h
= π×(2 in)2×3 in = 12π in3
10-A
Practice with
Surface Area and Volume Formulas
59/585 The water reservoir for a city is shaped like a
rectangular prism 300 meters long, 100 meters wide, and 15
meters deep. At the end of the day, the reservoir is 70% full.
How much water must be added overnight to fill the reservoir?
Volume of reservoir = 300 x 100 x 15 = 450000 cubic meters
30% of volume of reservoir has evaporated.
.30 x 450000 = 135000 cubic meters have evaporated.
135000 cubic meters must be added overnight.
10-A
Homework
Pages 589-590
# 34, 52, 58, 61, 84
Section 10B
Problem Solving
with
Geometry
pages 597-608
Pythagorean Theorem
For a right triangle with sides of length
a, b, and c in which c is the longest side
(or hypotenuse), the Pythagorean
theorem states:
a2 + b2 = c2
c
a
b
Pythagorean Theorem
example If a right triangle has two sides of lengths 9 in
and 12 in, what is the length of the hypotenuse?
(9 in)2+(12 in)2 = c2
81 in2+144 in2 = c2
225 in2= c2
225in  c
9
c
2
15in = c
12
Pythagorean Theorem
example If a right triangle has a hypotenuse of length 10
cm and a short side of length 6 cm, how long is the other
side?
(6)2 + b2 = (10)2
36 + b2 = 100
b2 = (100-36) = 64
b  64
b = 8 cm
10
6
b
Pythagorean Theorem
ex5/597 Consider the map in Figure 10.30, showing several city
streets in a rectangular grid. The individual city blocks are 1/8 of
a mile in the east-west direction and 1/16 of a mile in the northsouth direction.
a) How far is the library from the subway along the path shown?
b) How far is the library from the subway “as the crow flies”
(along a straight diagonal path)?
subway
library
Similar Triangles
Two triangles are similar if they have the same shape (but not
necessarily the same size), meaning that one is a scaled-up or
scaled-down version of the other.
For two similar triangles:
• corresponding pairs of angles in each triangle are equal.
Angle A = Angle A’, Angle B = Angle B’, Angle C = Angle C’
•the ratios of the side lengths in the two triangles are all equal
a b c
 
a' b' c'
B
a
A
c
B’
a’
C
b
A’
c’
b’
C’
Similar Triangles
67/605 Complete the triangles shown below.
50
x
60
y
10
40
Homework
Pages 603-605
#52,66,84,86,88