Section 10A Fundamentals of Geometry

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Section 10AFundamentals of Geometry

• Pages 578-588

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Perimeter and Area - Summary
10-A
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Perimeter and AreaRectangles
10-A
Perimeter l w l w 2l
2w Area length width l w
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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
• 33/589 A circle with diameter 25 centimeters
• Circumference pd p25 cm 25p cm
• Area pr2 p(25/2 cm)2 156.25p cm2

25
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Practice with Area and Perimeter Formulas
10-A

• 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

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2
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Practice with Area and Perimeter Formulas
10-A

• Find the circumference/perimeter and area for
  each figure described
• 37/589 A square with sides of length 12 miles
• Perimeter 12 km4 48 miles
• Area (12 meters)2 144 miles2

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12
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Practice with Area and Perimeter Formulas
10-A

• 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
• Area 20ft 5 ft 100 ft2

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10
5
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Practice with Area and Perimeter Formulas
10-A

• 45/589 Find the perimeter and area of this
  triangle
• Perimeter 9915 33 units
• Area ½ 154 30 units2

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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?
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Surface Area and Volume
10-A
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10-A
Practice with Surface Area and Volume Formulas

• 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.

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

• ex6/585 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 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
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.
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10-A

• Homework
• Pages 589-590
• 34, 52, 58, 61, 84

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Section 10BProblem Solvingwith Geometrypages
597-608
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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

• ex5/597 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
  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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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 603-605 52,66,84,86,88