Section 10A Fundamentals of Geometry

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Section 10AFundamentals of Geometry

• Pages 604-620

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Perimeter and Area - Summary
10-A
3
Perimeter and AreaRectangles
10-A
Perimeter l w l w 2l
2w Area length width l w
4
Perimeter and AreaSquares
10-A
Perimeter llll
4l Area length width l l
l2
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Perimeter and AreaTriangles
10-A
Perimeter a b c Area ½bh
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Perimeter and AreaParallelograms
10-A
Perimeter l w l w 2l
2w Area length height lh
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Perimeter and AreaCircles
10-A
Circumference(perimeter) 2pr
pd Area pr2 p 3.14159
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Practice with Area and Perimeter Formulas
10-A

• Find the circumference/perimeter and area for
  each figure described
• 43/617 A circle with diameter 16 centimeters
• Circumference pd p16 cm 16p cm
• Area pr2 p(16/2 cm)2 64p cm2

16
9
Practice with Area and Perimeter Formulas
10-A

• Find the circumference/perimeter and area for
  each figure described
• 51/617 A rectangular postage stamp with a length
  of 2.2 cm and a width of 2.0 cm
• Perimeter 2.2cm 2.2cm 2.0cm 2.0cm 8.4cm
• Area 2.2 cm 2.0 cm 4.4 cm2

2.2
2
10
Practice with Area and Perimeter Formulas
10-A

• Find the circumference/perimeter and area for
  each figure described
• 47/617 A square state park with sides of length 9
  miles
• Perimeter 9 mi4 36 miles
• Area (9 mi)2 81 miles2

9
9
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Practice with Area and Perimeter Formulas
10-A

• Find the circumference/perimeter and area for
  each figure described
• 49/617 A parallelogram with sides of length 12 ft
  and 30 ft and a distance between the 30 ft sides
  of 6 ft.
• Perimeter 12ft 30 ft 12ft 30ft 84ft
• Area 30ft 6 ft 180 ft2

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12
6
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Practice with Area and Perimeter Formulas
10-A

• 55/617 Find the perimeter and area of this
  triangle
• Perimeter 558 18 units
• Area ½ 83 12 units2

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Applications of Area and Perimeter Formulas
57/617 A picture window has a length of 8 feet
and a height of 6 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?
59/618 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 11 feet tall. What is the area of the
region to be covered in that case?
61/618 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?
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Surface Area and Volume
10-A
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10-A
Practice with Surface Area and Volume Formulas

• 89/619 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.

SoftballSurface Area 4xpx(2)2 16p square
inchesVolume (4/3)xpx(2)3 (32/3) p cubic
inches
Bowling ballSurface Area 4xpx(6)2 144p
square inchesVolume (4/3)xpx(6)3 288 p cubic
inches
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10-A
Practice with Surface Area and Volume Formulas

• ex6/613 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 pr2h p(1.5 in)24 in 9p
in3 Volume Can 2 pr2h p(2 in)23 in
12p in3
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Practice with Surface Area and Volume Formulas
69/618 The water reservoir for a city is shaped
like a rectangular prism 250 meters long, 60
meters wide, and 12 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 250 x 60 x 12 180000
cubic meters
30 of volume of reservoir has evaporated. .30 x
180000 54000 cubic meters have
evaporated. 54000 cubic meters must be added
overnight.
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10-A

• Homework
• Pages 617-618
• 46,52,54,58,62,68,71

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Section 10BProblem Solvingwith Geometrypages
621-637
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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
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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 in2144 in2 c2 225
in2 c2
c
9
12
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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
10
6
b
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Pythagorean Theorem

• ex5/626 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
  north-south direction.
• How far is the library from the subway along the
  path shown?
• How far is the library from the subway as the
  crow flies (along a straight diagonal path)?

subway
library
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ex6/626 Find the area, in acres, of the mountain
lot shown below.
250 ft
1200 ft
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Optimization
ex9/629 You have 132 meters of fence that you
plan to use to enclose a corral on a ranch. What
shape should you choose if you want the corral to
have the greatest possible area? What is the
area of this optimized corral?
87/634 Suppose you work for a company that
manufactures cylindrical cans. Which will cost
more to manufacture a can with a radius of 4
inches and a height of 5 inches or a can with
radius 5 inches and a height of 4 inches? Assume
the cost of material for the tops and bottoms is
1.00 per square inch and the cost of material
for the curved surface is 0.50 per square inch.
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Optimization
101/635 Telephone cable must be laid from a
terminal box on the shore of a large lake to an
island. The cable costs 500 per mile to lay
underground and 1000 per mile to lay underwater.
(See Figure 10.40/635) . As an engineer on the
project, you decide to lay 3 miles of cable along
the shore underground and then lay the remainder
of the cable along a straight line underwater to
the island. How much will this project cost?
Your boss examines your proposal and asks whether
laying 4 miles of cable underground before
starting the underwater cable would be more
economical. How much would your bosss proposal
cost? Will you still have a job?
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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

B
a
b
A
C
c
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Similar Triangles
67/605 Complete the triangles shown below.
50
x
y
10
60
40
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Homework Pages 633-635 70, 88, 94, 96